Simple physical constraints in hemolysis

J. theor. Biol. (1995) 175, 517–524

Simple Physical Constraints in Hemolysis M D. D

Physics Department, Hofstra University, Hempstead, New York, NY 11550, U.S.A.

(Received on 6 July 1994, Accepted in revised form on 3 April 1995)

The percentage of normal, red blood cells that are hemolyzed when placed in hypotonic solutions depends on a variety of factors, two important ones being the initial sphericities of the cells and the tonicities to which they are subjected. Other, less well-understood factors that are important in hemolysis are the initial cell volumes, how much free water they contain and the elasticity of the cell membranes. The purpose of this work is to identify the constraints a red cell must satisfy in order to be hemolyzed. Human erythrocyte data is used in a physical model that compares the balance of hydrostatic stresses in sphered cells that are on the verge of hemolysis. For hemolysis to occur we find there is a critical sphericity index that must be exceeded. It depends on tonicity, the initial, fractional water volume in the cells and the maximum fractional area dilation the cell membranes can withstand. Membrane tensile strength and the non-ideal osmotic behavior of hemoglobin are of relatively minor importance. But when they are taken into account, the hemolysis constraint, in the form of a remarkably simple inequality, compares favorably with clinical tests of erythrocyte osmotic fragility. 71995 Academic Press Limited

thin, condensed, liquid-like, very nearly unstretchable lipid bilayer that is linked (via integral proteins) to and supported by a highly flexible, very stretchable, protein lattice—the so-called membrane skeleton or cytoskeleton. The elastic properties of this laminar structure have been measured and may be used to evaluate the relative importance of different terms in the critical conditions for osmotic rupture of red cells. The critical conditions or constraints can be obtained from a very simple, well-known analysis of the basic physics of a spherical cell that is just on the verge of hemolysis: the transmembrane pressure difference must be sufficient to halt the further osmosis of water into the maximally swollen cell. By examining the necessary volume of water that must enter an initially discoid erythrocyte in order to swell it to a maximal spherical volume, we show that the fate of a red cell placed in a hypotonic solution largely depends on tonicity, the initial sphericity of the cell, the initial fraction of its volume not occupied by water, and, equally important, the maximum fractional amount by which its surface area can be stretched. Interestingly, the measured value for membrane tensile strength

1. Introduction A normal distribution of human red blood cells (RBCs) suddenly placed in distilled water will swell from their usual discoid shapes to spheres. At some critical point in the sphering process hemoglobin molecules escape across the cell membrane into the external water. This well-known hemolysis phenomenon will be incomplete if the external water contains as little as 0.4% of dissolved NaCl. Cells that are not hemolyzed do not meet or exceed some critical set of physical conditions, which presumably depends on the maximum amount of water the cells can absorb before the rising pressure in their interiors overwhelms the ultimate tensile strength of their membranes. The actual dynamics of the disc-to-sphere transformation are unknown, but extensive studies of red cell geometry and deformability allow the spherical end-state of this transformation to be described in elementary terms. Over the past 20 years a simplified working model for the erythrocyte membrane has emerged (Mohandas & Evans, 1994). The model considers a 0022–5193/95/160517+08 $12.00/0

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indicates that it contributes only very minor resistance to the osmotic rupture of erythrocytes. The fact that the osmotic coefficient of hemoglobin decreases as its concentration decreases can easily be incorporated into the hemolysis model examined here. It too has only a minor effect on osmotic fragility. With or without these corrections, the hemolysis constraint can be written in the form of a critical sphericity index (SI)c which must be exceeded if a cell is to be osmotically ruptured in a hypotonic solution. Using measured parameters for normal (and abnormal) human erythrocytes and assuming a Gaussian distribution of sphericity indices, the fraction of any sample of cells hemolyzed at different solution tonicities can be calculated and compared to clinical tests of osmotic fragility. The model provides a quantitative basis for interpreting these tests.

it swells to a spherical shape with an enlarged surface area A2=A1 (1+a),

(6)

where a=(A2−A1 )/A1 is the fractional area dilation. The sphered RBC will have a radius r=(A2 /4p)1/2 and a volume V2=4p(A2 /4p)3/2/3. Or, using eqns (5) and (6), r=(3V1 /4p)1/3(1+a)1/2/(SI)1/2 ,

(7)

2. The Hemolysis Model

V2=V1 (1+a)3/2/(SI)3/2 .

(8)

Consider a spherical cell of radius r, wall thickness t, interior pressure Pi , exterior pressure Po , interior impermeant solute concentration [C]i , and exterior impermeant solute concentration [C]o . For rwt and PiqPo , the average, circumferential, or hoop stress S in the cell wall needed to maintain hydrostatic equilibrium is given by

If f is the initial fraction of the cell volume not occupied by water (e.g. taken up by hemoglobin molecules) then the initial H2 O volume in the cell will be

and

St=r(Pi−Po )/2,

(1)

where St is simply the isotropic surface tension in the cell membrane. If the breaking stress (ultimate tensile strength) for the cell wall is SB , then the cell will rupture (hemolyze) when SqSB , or: r(Pi−Po )q2tSB .

(2)

The transmural pressure difference needed to halt the osmosis of water into the cell is given by van’t Hoff’s Law Pi−Po={fi [C]i−fo [C]o }RT,

(3)

where fi and fo are osmotic coefficients, T is the absolute temperature and R is the universal gas constant. The condition for cell rupture becomes r{fi [C]i−fo [C]o }q2tSB /(RT).

(4)

Consider a discoid RBC in an isotonic solution. In its interior let the impermeant solute concentration be [C1 ]i . With volume V1 and surface area A1 it has a sphericity index (SI)=(36p)1/3(V1 )2/3/A1 .

(5)

(Note that for a sphere (SI)=1.) Suppose the cell is placed in a hypotonic solution such that, by osmosis,

(VH2 O )1=V1 (1−f ).

(9)

If, by osmosis, the discoid swells to the spherical volume V2 , then the new H2 O volume in the cell will be (VH2 O )2=V2−fV1=V1 [(1+a)3/2/(SI)3/2−f ]. (10) In terms of the initial concentration [C1 ]i , the new, diluted concentration is [C2 ]i=[C1 ]i (VH2 O )1 /(VH2 O )2 , or, using eqns (9) and (10), [C2 ]i=

[C1 ]i (1−f ) . [(1+a)3/2/(SI)3/2−f ]

(11)

At the point of rupture, when the area dilation a is the maximum possible, this concentration will be the minimum possible. The sphered radius r [eqn (7)] will be the maximum possible when cell rupture occurs. Using these in eqn (4), the condition for cell rupture may be written (1−f ) f [C] − o o [(1+a)3/2/(SI)3/2−f ] fi [C1 ]i −

2tSB (SI)1/2/(1+a)1/2 q0. fi [C1 ]i RT(3V1 /4p)1/3

(12)

Equation (12), which can be algebraically manipulated into many different forms, embodies the simplest physical constraints a RBC must satisfy in order for it to be hemolyzed. It is a necessary and

   

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sufficient condition for hemolysis of a RBC placed in a hypotonic solution. In its present form it involves the sum of three, dimensionless, terms, each of which has a different dependence on the initial state of the cell. The first term is entirely geometrical since it does not depend on the physical dimensions of the cell, the temperature or the intracellular concentration of impermeant solutes. The second term is simply a measure of the tonicity of the external solution to which the cell is subjected. The third term is physical because it does depend on the cell dimensions, temperature and impermeant solute concentration: it essentially compares the osmotic pressure inside the cell to the maximum stresses the cell membrane can withstand. To evaluate the relative importance of these terms in determining the conditions for hemolysis we must use reliable data for RBCs.

s={[1/(SAI)2 ]−(SI)2}1/2 . Using a specific density for erythrocytes of 1.08, Freedman & Hoffman (1979) determined f=0.283, the fractional cell volume not occupied by H2 O, by weighing packed RBCs before and after desiccation. They also determined the changes in the osmotic coefficient for hemoglobin as a function of its concentration. The initial intracellular concentrations [C1 ]i=289.3 Mm−3, [Hb1 ]= 7.3 Mm−3 and their osmotic coefficients f1i=0.971, f1Hb=2.86 were taken from the metabolic model of Joshi & Palsson (1989). The average osmotic coefficient fo=0.93 assumes the external solution is composed of NaCl dissolved in H2 O. For physiological conditions T=310 K, and the universal gas constant is R=8.31 J(M.K)−1 . The final, hemoglobin osmotic coefficient f2Hb=1.587 and the final average intracellular osmotic coefficient f2i=0.939 (for maximally swollen cells) are calculated in the next section.

3. Human Erythrocyte Data

4. Approximations, Corrections and Predictions of the Model

Table 1 lists the pertinent RBC parameters we shall use in evaluating the relative importance of the three terms in the hemolysis condition of eqn (12). Linderkamp and Meiselman (1982) measured the cell volumes and surface areas of 200 RBCs using micropipet aspiration and vieomicrographs: their results compare favorably with other methods (Hochmuth, 1987; Fung, 1981). The mean sphericity index (SI) in Table 1 was calculated from Linderkamp’s and Meiselman’s measurements for the mean surface area index ((SAI)=1.3920.09) under the assumption that (SAI), which is the reciprocal of (SI) is normally distributed (i.e. (SI)={1/(SAI)}). The standard deviation for (SI), was calculated by taking the square root of the variance:

Denoting the third, or physical, term in eqn (12) by e, e=

2tSB (SI)1/2/(1+a)1/2 , fi [C1 ]i RT(3V1 /4p)1/3

(13a)

if we use the RBC values from Table 1, then e=8.61×10−3 .

(13b)

For a typical RBC (mean values in Table 1) the geometrical, or first term in eqn (12) is on the order 0.505, so the physical term, which depends on the membrane tensile strength (tSB ), is a rather minor correction (1.7%) in comparison. If we neglect it (which is tantamount to assuming tSB=0), the

T 1 Model parameters for human erythrocytes Parameter

Symbol

Mean, initial cell volume Mean, initial cell surface area Mean sphericity index Sphericity, Std. dev. Maximum, cell surface tension Maximum, fractional area dilation Initial, fractional cell volume not occupied by water Hemoglobin, osmotic coefficient Extracellular solute concentration Average extracellular osmotic coefficient Initial, intracellular solute concentration Initial, average, intracellular osmotic coefficient Initial, Hemoglobin concentration Initial, Hemoglobin osmotic coefficient Final, Hemoglobin osmotic coefficient Final, average, intracellular osmotic coefficient

V1 A1 (SI) s tSB a f

Value =(8.9821.27)×10−17 m3 =(1.34120.138)×10−10 m2 =0.722 =0.047 =10−2 Nm−1 =0.03 =0.283

fHb =1+0.0645[Hb]+0.0258[Hb]2 [C]o E289.3 Mm−3 fo =0.93 [C1 ]i =289.3 Mm−3 f1i =0.971 [Hb1 ] =7.3 Mm−3 f1Hb =2.86 f2Hb =1.587 f2i =0.939

Reference Linderkamp & Meiselman (1982) Linderkamp & Meiselman (1982) (See text) Linderkamp & Meiselman (1982) (See text) Linderkamp & Meiselman (1982) Mohandas & Evans (1994) Evans et al. (1976) Freedman & Hoffman (1979) Freedman & Hoffman (1979) Adjustable (See text) Assumed (See text) Joshi & Palsson (1989) Joshi & Palsson (1989) Joshi & Palsson (1989) Joshi & Palsson (1989) Calculated (See text) Calculated (See text)

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hemolysis constraint, eqn (12), takes the particularly simple form (SI)q(SI)c=

(1+a) . { f+(1−f )(fi [C1 ]i )/(fo [C]o )}2/3 (14a)

That is, there is a critical sphericity index (SI)c which an erythrocyte must exceed if it is to be hemolyzed when placed in a hypotonic solution. To a first approximation, (SI)c only depends on tonicity (fo [C]o /fi [C1 ]i ), the erythrocyte’s fractional volume f not occupied by water and the maximum fractional dilation a of its surface area. If we include the physical term e (i.e. tSBq0), the hemolysis constraint has the more complicated form of (SI)q(SI)c= (1+a){1+e(fi [C1 ]i )/(fo [C]o )}2/3 . {f+(1−f )(fi [C1 ]i )/(fo [C]o )+ef(fi [C1 ]i )/(fo [C]o )}2/3 (14b) The average osmotic coefficient fi in eqn (14b), or its approximate version [eqn (14a)], is the final coefficient when the sphered cell is on the verge of rupture. It would be equal to the initial average coefficient f1i , before swelling, if the impermeant solutes behaved ideally. However, because of the large size difference between hemoglobin and water molecules, the hemoglobin osmotic coefficient exhibits a strong concentration dependence. Using data for ox and sheep hemoglobin, Freedman & Hoffman (1979) have experimentally fit this dependence by the following virial equation: fHb=1+0.0645[Hb]+0.0258[Hb]2.

(15)

Consequently, as an erythrocyte swells from a discoid to a sphere, [Hb] will decrease and so will the values for fHb and fi . The dilution of [Hb] is described by eqn (11): [Hb2 ]=

[Hb1 ](1−f ) , {(1+a)3/2/(SI)3/2−f }

(16)

where [Hb1 ] and [Hb2 ] are, respectively, the initial and final hemoglobin concentrations. The final concentration in any particular cell depends on its specific values for (SI), a and f. We may estimate it for a typical erythrocyte by using the mean values from Table 1 in eqn (16): [Hb2 ]=0.5046[Hb1 ]= 3.684 Mm−3. Substituting in eqn (15), the final hemoglobin osmotic coefficient, for a typical RBC, is f2Hb=1.587,

(17)

which is lower than the initial value f1Hb=2.86. Since, during swelling, the amount of hemoglobin and other impermeant solutes does not change, the final average osmotic coefficient f2i is related to the initial value f1i by fi=f2i=f1i−(f1Hb−f2Hb )[Hb1 ]/[C1 ]i .

(18a)

Thus, for a typical erythrocyte f2i=0.939.

(18b)

The major prediction of the model is that there is a critical sphericity index (SI)c that must be exceeded if a RBC is to be hemolyzed. For zero tensile strength membranes and constant osmotic coefficients (fi=f1i ), the dependence of (SI)c on the external concentration [C]o is given by eqn (14a). When the tensile strength of the membrane and non-ideal osmotic behavior are taken into account, the dependence of (SI)c on [C]o is given by eqn (14b). However, according to eqn (13a) for e and eqns (15), (16) and (18a) for fi=f2i , these corrections depend on (SI)c . As a starting approximation, one can use the estimates e=8.61×10−3 and f2i=0.939 in eqn (14b) and then calculate (SI)c for each [C]o . Or, starting with these estimates, one can numerically iterate eqns (14b), (16), (15) and (18a) to obtain the exact values for e, fi and (SI)c at each external solution concentration [C]o . For concentrations in the range 0.25E%NaClE 0.70, the exact values for (SI)c are about 3% larger than those predicted by eqn (14a). The results are shown in Fig. 1. It is interesting to note that membrane tensile strength and the non-ideal osmotic behavior of hemoglobin confer a small, but measurable, degree of osmotic stability to RBCs. This is apparent in Fig. 1: if all erythrocytes had the same sphericity index (SI)=0.722, idealized erythrocytes with zero tensile strength membranes and osmotically ideal solutes (broken curve) would hemolyze in hypotonic solutions with %NaClE 0.442, while the realistic erythrocytes described by eqn (14b) (solid curve) would hemolyze at the lower tonicity %NaClE0.423. The non-ideal osmotic corrections are responsible for about two-thirds of this small effect, the remainder being attributable to the finite tensile strength of the membrane. 5. The Osmotic Fragility Test The human erythrocyte osmotic fragility test is a fairly routine diagnostic procedure in clinical medicine (Brown, 1988). It provides a convenient experimental test of the validity of the hemolysis model presented here. In any sample of RBCs there will be a distribution of sphericity indices; those cells with the

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(SI)=0.722 and s=0.047 and the erythrocyte model parameters listed in Table 1 the results are shown in Fig. 2. The shaded region displays the range of osmotic fragility curves (% hemolysis or F) expected for normal blood (Brown, 1988). The broken curve is for the idealized erythrocytes described by eqn (14a) and the solid curve is for the realistic erythrocytes described by eqn (14b). The general shape of these two curves is similar to the envelopes of the ‘‘normal’’ range, and it is again interesting to note that the small corrections for membrane tensile strength and non-ideal osmotic behavior give near perfect agreement with clinical expectations for normal RBCs. In view of some of the uncertainties in the human erythrocyte data listed in Table 1 we take this as reasonable confirmation of the hemolysis model presented here. The osmotic fragility test is essentially a measure of spherocytosis; it gives a more objective measurement than visual inspection of a blood smear. While it can be interpreted in terms of some ‘‘normal’’ critical sphericity index, it is incapable of saying what this index is or on what it depends. As such, it is still a qualitative test. The hemolysis model presented

0.90

Critical sphericity index (SI )c

0.85

0.80

0.75

(SI ) = 0.722

0.70

0.65

0.60

0.55

0.50

0.6

0.5

0.4

521

0.3

% NaCl F. 1. Critical sphericity index as a function of extracellular solute concentration. The solid curve includes corrections for non-ideal osmotic behavior and finite membrane tensile strength. The broken curve neglects these corrections.

1.0

highest indices (most spherical) are at greatest risk for hemolysis. According to eqn (14a) or (14b), all cells with sphericity indices greater than (SI)c will be hemolyzed. To obtain the fraction F hemolyzed at a particular solution tonicity, we assume a reasonably narrow, normalized Gaussian distribution of sphericity indices and integrate over (SI), from (SI)c to infinity. Note, the error in letting (SI)q1 in the integration is minimal for a narrow distribution (small standard deviation). The result can be written in terms of the error function Erf(x) (Abromowitz & Segun, 1968):

0.8

F=

6

7

1 x 1− Erf(=x =/(21/2s)) , 2 =x =

(19)

0.9

0.7

F

0.6 0.5 0.4 0.3 0.2 0.1

where x=(SI)c−(SI).

(20)

(SI)c , which depends on fo [C]o , is calculated from either eqn (14a) or (14b), depending on the accuracy desired. By varying fo [C]o we may compute F as a function of [C]o for any choice of model parameters. For

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

% NaCl F. 2. Osmotic fragility curves. Fraction F hemolyzed as a function of external solute concentration. The shaded region indicates normal clinical range. The solid curve is for model parameters in Table 1. The broken curve neglects osmotic and tensile-strength corrections.

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T 2 Model parameters in hereditary spherocytosis Parameter Mean, initial cell volume Mean sphericity index Sphericity, Std. dev. Maximum, cell surface tension Maximum, fractional area dilation Mean corpuscular Hemoglobin concentration Initial, fractional cell volume not occupied by water water Initial, Hemoglobin concentration Initial, Hemoglobin osmotic coefficient Initial, intracellular solute concentration Initial, average intracellular osmotic coefficient Extracellular solute concentration Average extracellular osmotic coefficient

Symbol

Value

Reference

V1 (SI) s tSB a MCHC f

=8.00×10−17 m3 =0.792 =0.0765 =10−2 Nm−1 =0.03 =38 gdl−1RBC =0.319

Williams et al. (1990) Best fit to data from Cooper & Bunn (1977) Best fit to data from Cooper & Bunn (1977) Mohandas & Evans (1994) Evans et al. (1976) Cooper & Bunn (1977) Estimated (see text)

[Hb1 ] f1Hb [C1 ]i f1i [C]o fo

=8.65 Mm−3 =3.49 =278.2 Mm−3 =1.01 E289.3 Mm−3 =0.93

Estimated (see text) Estimated (see text) Estimated (see text) Estimated (see text) Adjustable (see text) Assumed (see text)

here is potentially useful because it provides a broader, more quantitative way of interpreting the fragility curves in terms of the model parameters. An abnormal fragility curve would suggest an abnormality in one or more of the erythrocyte parameters listed in Table 1. The fragility curve alone, however, is not sufficient to determine where the abnormalities lie; other clincial tests must be made to assess the state of a person’s erythrocytes. If we use the model parameters in Table 1 as prototypical for erythrocytes, then it is shown in the Appendix how clinical measurements for mean corpuscular volume, MCV, in fl, and mean corpuscular hemoglobin concentration, MCHC, in grams per deciliter of packed red cells (g dl−1 RBC), can be used to generate the person’s model parameters for V1 , [Hb1 ], f, f1Hb , [C1 ]i and f1i . These new model parameters can then be used to assess the abnormalities in a person’s osmotic fragility curve, as we now show.

6. Osmotic Fragility in Hereditary Spherocytosis In hereditary spherocytosis (HS), it is believed (Mohandas & Evans, 1994; Palek, 1990) that membrane mechanical instabilities due to deficiencies in spectrin (and possibly other proteins) in the cytoskeleton lattice lead to lipid bilayer loss. The resulting cells tend to be smaller and more spherical than normal, with elevated hemoglobin concentrations. The disease can vary from mild to life-threatening, the most severe cases having large populations of hyperchromic cells lacking central depressions. Accordingly, the osmotic fragility test for this disorder can vary from near normal to highly abnormal. In the most common form of this anemia, mean corpuscular volume is on the low side of normal (MCV=80.0 fl) and the mean corpuscular hemoglobin

concentration is elevated to MCHC= 38 gdl−1 RBC (Cooper & Bunn, 1977). Using these clinical values in eqns (A.1)–(A.9) in the Appendix, we have the following erythrocyte model parameters for hereditary spherocytosis: V1=8.00×10−17 m3, f=0.319, [Hb1 ]=8.65 Mm−3 , f1Hb=3.49, [C1 ]i=278.2 Mm−3 , and, f1i=1.01. The disparity between the normal RBCs (as described by the parameters in Table 1) and HS cells is worth commenting on: HS cells are smaller with a greater fraction of their volumes not occupied by H2 O. Their hemoglobin concentrations are about 18.5% greater than normal and their intracellular impermeant solute concentrations are low by 4%, but have an abnormally high average osmotic coefficient. Since the defect in HS is not believed to directly involve the elastic properties of the lipid bilayer, which offers virtually all the resistance to membrane surface area dilation, we shall take the following values from Table 1 as representative of HS cells: tSB=10−2 Nm−1 a=0.03. Missing from our list of HS model parameters, but included in Table 2, is the mean sphericity (SI) and

    1.0

7. Discussion

0.9 0.8 0.7

F

0.6 0.5 0.4 0.3 0.2 0.1

0.8

523

0.7

0.6

0.5

0.4

0.3

0.2

0.1

% NaCl F. 3. Osmotic fragility curve in Hereditary Spherocytosis. Data points taken from Cooper & Bunn (1977). The solid curve is best fit using model parameters in Table 2: (SI)=0.792, s=0.0765. The shaded region indicates clinical range expected for normal erythrocytes.

its standard deviation s. These may be obtained by finding the best fit to an actual HS osmotic fragility curve. That is, we use the above model parameters in eqns (14)–18), of Section 4, and eqns (19) and (20), of Section 5, while using different trial values for (SI) and s to generate many different, theoretical, osmotic fragility curves. The curve that best fits the experimental HS fragility curve yields the HS model values for (SI) and s. The best fit to the data taken from Cooper and Bunn (1977) is shown in Fig. 3. The shaded region shows the normal clinical range. The fit was obtained for (SI)=0.792 and s=0.0765, which indicates that this is a broader distribution of cells that are somewhat more spherical than normal. The fact that the model is consistent with clinical measurements is again taken to be reasonable confirmation of its validity. Importantly, the quantitative value of the model in interpreting the fragility curve is quite evident for here it yields meaningful numbers which clearly indicate that this distribution of cells is abnormal.

In summary, we have described a physical model for red cell hemolysis. An experimentally based set of data for normal human erythrocytes, which is independent of the model, has been used to evaluate the relative importance of different factors in the hemolysis of red cells. By examining the balance of hydrostatic stresses in a sphered RBC that is on the verge of hemolysis, it has been shown that there is a critical sphericity index for hemolysis that is only weakly dependent on erythrocyte membrane tensile strength and the non-ideal osmotic behavior of hemoglobin. We have shown that the physical constraints in hemolysis reduce to an inequality among geometrical parameters for RBCs and the solution tonicities they are placed in. When corrections for tensile strength and non-ideal osmotic behavior are included, the model is consistent with clinical osmotic fragility tests. It provides a quantitative basis for interpreting these tests. From routine clinical measurements on a person’s blood, we have shown how to generate erythrocyte parameters that are specific for that person. For the particular case of hereditary spherocytosis, when the appropriate parameters are used in the hemolysis model, it yields an osmotic fragility curve that is consistent with the clinical curves in this disorder; it also completes the profile of the cells in HS by indicating their mean sphericities and how broadly these sphericities are distributed.

I thank Mr. Joe Howard, Dr. Man S. Oh and Dr. Barbara Delano for informative discussions.

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APPENDIX Clinical Measurements for Erythrocyte Model Parameters The erythrocyte model parameters in Table 1 were pooled from the measurements of many separate researchers using blood samples from an even larger number of donors. As such they do not represent the parameters for any one person, but because of their inner consistency and success in predicting the normal osmotic fragility curve, they may be treated as the prototypical parameters for all normal erythrocytes. Many of these parameters correspond to a particular set of normal values in some of the fairly routine clinical measurements made on human blood. Making some simplifying assumptions, they may be used to generate another set of normal (or abnormal) parameters from any other particular set of clinical measurements. For example, a RBC count and hematocrit will yield a mean corpuscular volume MCV which is related to the mean cell volume V1 by: V1=MCV×10−18 m3,

(A.1)

where MCV has the normal range 80.0 to 96.1 fl (for V1=8.98×10−17 m3, in Table 1, this would correspond to a normal MCV=89.8 fl). Another clinical measurement, the mean corpuscular hemoglobin (MCH), is related to the number of moles of hemoglobin MHb in an average red cell according to: MHb=1.5504×10

−17

(MCH) M,

(A.2)

where MCH, in picograms per red cell, has the normal range MCH=27.5 to 33.2 pg/RBC (Williams et al., 1990). Sometimes the mean corpuscular hemoglobin concentration (MCHC) is measured: MCHC=100 MCH/MCV, (A.3) where MCHC has the normal range 33.4 to 35.5 gdl−1RBC (Williams et al., 1990). However, this

clinical measurement does not specify the actual concentration [Hb] in the cell. For a RBC with volume V1 and MHb moles of hemoglobin, only the fractional volume (1−f ) is occupied by water so the concentration inside the cell is: [Hb]=MHb /{(1−f)V1 }.

(A.4)

(From Table 1, [Hb]=7.3 Mm−3 (Joshi & Palsson, 1989), V1=8.98×10−17 m3 (Linderkamp & Meiselman, 1982) and f=0.283 (Freedman & Hoffman, 1979), this would mean a specific cellular hemoglobin content of MHb=4.7×10−16 M and a specific normal MCH=30.3 pg/RBC or a specific normal MCHC=33.8 gd l−1RBC. It is significant that separate model parameters from three different research groups combine to give an acceptable, normal clinical value.) If we make the simplifying assumption that the cell volume not occupied by water, fV1 , is proportional to the number of moles of Hb in the cell, fV1=bMHb , then from the values in Table 1, b=0.05407. Consequently, for any cell of volume V1 containing MHb moles of Hb the fractional volume not occupied by water may be estimated from f=0.05407MHb /V1 .

(A.5)

Once f is known, [Hb] may be calculated from eqn (A.4) and its osmotic coefficient determined from the Freedman–Hoffman equation: fHb=1+0.0645[Hb]+0.0248[Hb]2.

(A.6)

Presumably, the external milieu of any distribution of RBCs, normal or otherwise, is typical isotonic plasma whose osmolarity may be calculated from the values in Table 1: fi [C]i=0.971[289.3]= 280.9 osM m−3 . If we assume that all intracellular solutes, S, other than Hb, behave ideally with constant osmotic coefficients fs=0.93 then fHb [Hb]+0.93[S]i=280.9 osM m−3 .

(A.7)

Since the total intracellular concentration is [C]i=[Hb]+[S]i , we may use eqn (A.7) to solve for [C]i=302.04−[Hb](1.0753fHb−1) Mm−3,

(A.8)

and its average osmotic coefficient fi=280.9/[C]i .

(A.9)

Thus, the specific results of clinical measurements for MCV, and MCHC (or MCH) will, using eqns (A.1)–(A.9), generate a specific set of erythrocyte model parameters for V1 , f, [Hb1 ], f1Hb , [C1 ]i and f1i .