A model of microvascular oxygen transport in sickle cell disease

MICROVASCULAR

RESEARCH

30, 195-206 (1985)

A Model MARGARET

of Microvascular Oxygen Transport in Sickle Cell Disease’

M. VAYO, HERBERT H. LIPOWSKY, NOLAN KARP, EMILY SCHMALZER, AND SHU CHEIN

Department

of Physiology,

College of Physicians and Surgeons, Columbia New York, New York 10032

University,

Received April 25, 1984

The model of local control of oxygen delivery in the microvasculature developed by H. J. Granger and A. P. Shepherd (1973, Microvasc. Res. 5, 49-72) was extended to describe microcirculatory blood flow in sickle ceh disease. Two major characteristics of sickle cell blood were incorporated into the model: (1) an abnormal blood viscosity which is dependent on the degree of hemoglobin oxygen saturation and hematocrit, and (2) a reduced affinity of hemoglobin (Hb) for oxygen. Sickle cell blood viscosity as a function of oxygen saturation and hematocrit was modeled empirically based upon existing data. Alterations in HbOz affinity were studied in the model by introducing PsOas an independent variable. The altered oxygen supply/demand relationship in sickle cell disease was simulated following an increase in tissue metabolic demand and a decrease in arteriolar blood flow. The results were analyzed to evaluate the roles of the various rheological characteristics of sickle cell blood in affecting microcirculatory dynamics and tissue oxygen delivery. It was demonstrated that, within the hematocrit range of 20 to 45%, the elevation of PjO from 27 to 38 mm Hg in sickle cell blood is adequate to compensate for the diminished 0, content, despite an elevated blood viscosity, and maintain near normal tissue ~0,. o 1985 Academic Press, Inc.

INTRODUCTION Although the phenomena that lead to a crisis in sickle cell anemia have been studied extensively, the interrelated events are difficult to analyze and remain unpredictable insofar as they affect microvascular function. There are many properties of sickle cell blood that contribute to the clinical manifestations of the disease. These include: (a) the anemia, which limits the oxygen-carrying capacity of the blood; (b) a decreased affinity of hemoglobin S for oxygen; and (c) the increased rigidity and subsequent sickling of red blood cells upon hemoglobin oxygen desaturation, leading to elevated blood viscosity and increased peripheral resistance and ultimately vaso-occlusion. These characteristics of sickle cell blood have the greatest impact in the microcirculation, where oxygen is delivered to the tissue and the hemoglobin becomes desaturated. The passageways provided by the capillaries are the narrowest and ’ Supported by NIH Research Grants HL-16581 and HL-28381, and NRSA HL-07114 for M.M.V. and E.S.. and RCDA HL-00594 for H.H.L. 195 0026-286Z85 $3.00 Copyright Q 1985 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in U.S.A.

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ET AL.

hence are most sensitive to changes in red cell rigidity. While the anemia in sickle cell disease tends to counteract the elevation in blood viscosity attendant to the increased red cell rigidity, it also reduces the oxygen carrying capacity of the blood. When the latter is combined with a reduction in O2 affinity, a decrease in oxygen saturation may result in the microcirculation, leading to an enhancement of sickling and a rise in microvascular resistance. A computer model of the normal microcirculation has been developed by Granger and Shephard (1973) to simulate intrinsic microvascular regulation of oxygen delivery to tissue. This model incorporates variations in capillary density and arteriolar resistance as feedback mechanisms which govern the microvascular response to altered oxygen supply or demand. In the present study, this model has been modified to simulate the microcirculatory transport characteristics of sickle cell blood, taking into account the alteration in HbOz affinity and the dependence of blood viscosity on hematocrit and oxygen saturation in this disease. The relative roles of these factors in affecting microcirculatory hemodynamics and oxygen transport have been examined under the separate conditions of increased tissue metabolic demand and decreased blood flow, with the aim of elucidating the events that lead to a full-fledged crisis. METHODS The model of Granger and Shepherd (1973), which simulates the local microvascular response to alterations in oxygen supply and demand, is divided into three interrelated sectons: (1) convective transport of oxygen through the microvasculature and its diffusion throughout the tissue, (2) oxygen utilization by the mitochondria and metabolic feedback, and (3) control of the hemodynamic resistance by arteriolar dilation/constriction and variation in capillary density. In sections (1) and (2), the oxygen flux across the capillary wall and the oxygen uptake by the mitochondria are computed along with the capillary ~0, and tissue pOZ. A critical level of pOZ in the tissue is maintained via two feedback control mechanisms: (1) changes in capillary density, and (2) arteriolar dilatation or constriction, as given by the following two equations: RA

=

RAN

+

&

,-

(~o~cc,,

-

pOZR)

dt,

(1) (2)

where RA = arteriolar resistance,

N = fraction of capillaries which are perfused, PO&” = actual tissue partial pressure of oxygen, POZR= partial pressure of oxygen required by the tissue; this variable is controlled at a set point of 5 mm Hg, KR,KN = system “gain” constants which represent the sensitivity of the feedback response (Granger and Shepherd, 1973). The subscript “0” denotes the initial values for the resting state. The effect of varying the fraction of perfused capillaries on the resistance to blood flow at the

OXYGEN

TRANSPORT IN SICKLE CELL DISEASE

197

capillary level (Rc) is given by

Rc = U%,IW Go.

(3)

With these feedback equations, the capillary density and arteriolar resistance are used to compute blood flow, venous pressure and capillary resistance in the hemodynamics section. In the present study, the decreased Hb02 affinity and abnormal blood viscosity of sickle cell blood have been incorporated into the model as described in the following sections. Oxygen Affinity

of Sickle Hemoglobin

In anemia the oxygen carrying capacity of blood is reduced and there is usually a compensatory decrease in HbOz affinity (increase in P,,,) characterized by a rightward shift of the hemoglobin-oxygen saturation curve which favors the unloading of oxygen to tissues. In contrast to the Granger-Shepherd model, where the HbOz affinity is taken to be constant, the present simulation allows the P,, to vary through the introduction of a parameter, A, into the equation proposed by Granger and Shepherd (1973) to describe the HbO,? equilibrium in the oxygen transport and utilization section of the model (schematized in Fig. 1): POZcap

=

-A Ml - ~~O~l,,,/~~~l,,~“21.

(4)

where [02 Icapand [O,],, are the capillary and arterial oxygen concentrations, respectively. By assuming the hemoglobin in the arterial blood to be 100% saturated with oxygen, Eq. (4) can be written as: POZcap

=

-A ln[l - S,,p1’2],

(44

where S,, is the fractional HbOZ saturation in the capillary blood. Variations in A from the normal value of 20 (used in the original model) are inversely related

to the HbOz affinity and explicitly related to the P,-,,by: A = -P,,/ln[l

- (OS)“‘].

(5)

This yields a PsOof 24.6 mm Hg for normal blood, where A=20. Apparent

Viscosity

of Sickle Cell Blood

The apparent viscosity of normal AA blood decreases in anemia, but its value is insensitive to the degree of oxygen saturation (Usami et al., 1975b). In contrast, the apparent viscosity of sickle cell blood (SS) depends not only on the hematocrit level, but also on the degree of saturation of the hemoglobin. A quantitative relationship may be derived from the data of Chien et al., (1976) describing the effect of hematocrit on the apparent viscosity of sickle cell blood (shear rate = 208 set-‘) for different levels of O2 saturation, as summarized in Fig. 2. The family of curves shown can be described by a single equation: rlss

=

rip

e3.16

Hs-4.26

H=.

(6)

where qss (in cP) is the apparent viscosity of SS blood and qp (= 1.4 cP) is the plasma viscosity, S is the fractional HbOz saturation, and H is the fractional hematocrit. When the blood is fully saturated (S= l), qss = 1.4 e3.i6H and the apparent viscosity approaches that of AA blood. Equation (6) was incorporated

198

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ET AL.

Blood Flow Venous Volume Venous Pressure

capLo21

I

N

I

I METABOLIC

OXYGEN

TRANSPORT

AND UTILIZATION

l-G--l

BLOOD

VISCOSITY

FIG. 1. Systems diagram of modified portions of Granger-Shepherd model. qAA, viscosity of normal AA blood; qss, viscosity of SS blood; Cap, capillary; Art, arteriolar; A, affinity parameter; S, fractional saturation; j(H), functional dependence on hematocrit; FC, capillary resistance factor; FA, arteriolar resistance factor; RC, capillary resistance; RA, arteriolar resistance; RC’,RA’: capillary and arteriolar resistances before multiplication by factors FC and FA, respectively.

into the model to determine the rheological effects of anemia and desaturation on local hemodynamic and metabolic parameters, such as microvascular resistance and venous p02. The apparent viscosity of normal blood (Q,.,) is independent of O2 saturation, and its relation to hematocrit can be described by the equation: T)AA= r), (1 + 8.57 H*).

(7)

The ratio of r&n AA can be viewed as a relative rheological factor comparing sickle cell blood with normal blood. This relative rheological factor for the calculation of capillary resistance (FC) is: Fc = e’,‘“,“~;~~‘.

(8)

Since S = 1 in arteriolar blood, the relative rheological factor for the calculation of arteriolar resistance (FA) is: 3 16

FAE

e’

H

1 + 8.57 H*’

(9)

OXYGEN

20

TRANSPORT

IN

SICKLE

lict=45 r

9

L

CELL

o

DATA

-

EOUATION

(Chien

et al..

3.18 ‘Is3

15

q

1.4

199

DISEASE

n

1976)

3-4.28

I+*

e

-

\; 5 -

10 ::

_

F

5 -

oL~~~‘l”“~~‘~~‘~” .1

.2

.3

.4

s (FRACTIONAL

.5

.6

.7

nbo2

SATURATION)

.0

.S

1 .O

2. Apparent sickle cell blood viscosity (qss) vs fractional HbOz saturation (S) for different hematocrits (Hct). H= Hct/lOO. Solid lines are theoretical prediction from equation shown; data points (0) are laboratory determinations made on sickle cell blood. FIG.

The factors FC and FA were incorporated into the hemodynamics section of the model to modify capillary and arteriolar resistances, respectively (see Fig. 1). Arteriolar resistance was adjusted for altered rheological properties of blood by introducing the factor FA into Eq. (1) as a multiplication factor, viz:

RA = FA [ho + KRI f~oz,, - POZ,)dtl.

(10)

The capillary resistance was similarly modified by introducing the factor FC into Eq. (3) for capillary resistance:

Rc = FC [(No/N)&,I.

(11)

When FC or FA is equal to one, the computed capillary (R,-) or arteriolar (RA) resistance is identical to that of the original model. These modifications of the Granger-Shepherd model allow an assessment of the contributions of the properties of sickle cell blood to microcirculatory hemodynamics . RESULTS Computation of Altered Oxygen Affinity in Response to Anemia As a first step in exploring the effects of SS blood characteristics on oxygen delivery, the model was used to examine the magnitude of the shift in oxygen

200

VAYO ET AL.

affinity required to maintain adequate tissue oxygen delivery in the face of anemia. The solid line in Fig. 3 represents the P5,,dictated by the model simulation to maintain a tissue intracellular ~0, of 5 mm Hg and a capillary pOz of 40 mm Hg. In this example, the only modification of the Granger-Shepherd model is an altered HbOz affinity. (The effects of the abnormal sickle cell blood viscosity on oxygen delivery are incorporated in the next section.) At any point along the curve in Fig. 3, a normal hemodynamic environment exists in the tissue with normal blood flow, capillary and arteriolar resistances, and capillary density. It should be noted that in response to anemia, there is usually a compensatory decrease in Hb02 affinity (increase in P5,,)to maintain adequate tissue oxygenation. The inverse relationship between PSOand hematocrit found in laboratory determinations on blood samples obtained from normal subjects (de Furia et al., 1974, Miller et al., 1980) is shown in Fig. 3, as well as those from patients with irondeficient anemia (de Furia et al., 1974) and sickle cell anemia (Miller et al., 1980, Bromberg and Jensen, 1967). These laboratory data agree with the simulation curve for hematocrit values above approximately 25%. Below this hematocrit, the shift in affinity observed in patients’ blood is less than that predicted by the simulation to maintain tissue oxygenation, indicating that there is a limit for the extent to which the Hb02 affinity can be decreased to enhance oxygen unloading. Tissue Oxygen Delivery in Resting Condition The distinguishing features of sickle cell blood, namely anemia, increased blood viscosity, and reduced HbOz affinity, can be examined for their contribution to local tissue hemodynamics. Four parameters which are relevant to oxygen delivery to tissue are shown in Fig. 4 for the resting condition: capillary pOz, capillary oxygen concentration ([0,]), capillary resistance, and blood flow. For each pa50

MODEL

SIMULATION

CJ 83

NORMAL Fe-DEF.

ANEMIA

SICKLE

I

fl

\

40-

CELL

(da Furia et al.. 1974) (Miller et al.. 1080)

0

(de Furia

.

(Miller et al.. 1880) (Eromberg and Jensen.

et aL. 1974)

1987)

2 2 n% 30-

201

1

1

10

15

20

25

1

1

I

I

I

30

35

40

45

50

HEMATOCRIT

FIG. 3. Solid line: model simulation of Psorequired to compensate for a reduced hematocrit, in order to maintain capillary ~0, at 40 mm Hg and tissue ~0, at 5 mm Hg. Data points are actual laboratory values from humans.

OXYGEN

TRANSPORT IN SICKLE CELL DISEASE

201

rameter, four bars are presented. The unshaded bar illustrates the normal case (set as 1.0). The remaining three bars represent the stepwise additions of the characteristics of sickle cell blood: anemia alone, anemia combined with increased viscosity (due to desaturation of hemoglobin S), and anemia combined with increased viscosity and decreased PsO; all values have been normalized to the normal resting value. When anemia alone is present (Fig. 4, dotted bars), the reduction in red cell concentration is accompanied by an increase in blood flow which serves to maintain an adequate oxygen supply to the tissue. More capillaries are also opened (capillary density rises to 1.3 times the normal resting value), lowering the capillary resistance. With decreases in hematocrit and arterial [O,], the capillary oxygen concentration and capillary pOZ also decrease. The combination of anemia with the increase in blood viscosity of deoxygenated sickle cells (Fig. 4, cross-hatched bars) results in an increased capillary resistance and a reduced blood flow, in spite of an increased capillary density (1.6 times normal). The capillary oxygen concentration and capillary pOZ are further reduced. The superposition of the reduction of HbOz affinity upon anemia and elevated blood viscosity (Fig. 4, solid black bars) results in an increased oxygen extraction, a lowered capillary [O,], and an increased capillary ~0,. Thus the reduced HbOz affinity has a compensatory effect through restoration of the capillary ~0, and

UNSTRESSED

CONDITION

FIG. 4. Resting condition for capillary pOz, capillary [O,], capillary resistance, and blood Bow. Unshaded bars: AA blood (normal hematocrit, viscosity, and HbO, affinity); dotted bars; reduced hematocrit alone (22%); cross-hatched bars: reduced hematocrit and increased blood viscosity according to Eq. (6); solid black bars: reduced hematocrit, increased viscosity, and decreased HbO, affinity (P50 = 37 mm Hg). Value of each variable in resting condition for AA blood is used to normahze the results. Normal resting state vahres are: capillary resistance, 2.0 mm Hg/ml/min/lOt) g; blood flow, 10 ml/min/lOO g; capillary pOz, 40 mm Hg; capillary [O,], 15 ~01%.

202

VAYO

ET AL.

hence tissue ~0~. The rise in capillary ~0, leads to a decrease of capillary density toward normal (1.3 times normal), an increase in capillary resistance, and a reduction in blood flow. Effects of Increased

Metabolic

Demand on Oxygen Delivery

When the metabolic demand of the tissue is increased (e.g., during muscular exercise), local changes in the microvasculature occur to provide an appropriate oxygen supply. Two levels of increased metabolic demand ( + 25 and + 125%) were simulated, and the resulting changes in four variables of the model that reflect the state of the tissue capillary bed (i.e., the capillary ~0, and [O,], the capillary resistance, and the blood flow) are illustrated in Fig. 5. According to the Granger-Shepherd model for normal blood, an increase in oxygen demand by the tissue causes reductions in capillary pOz and [O,], which lead to capillary recruitment, thus decreasing the capillary resistance and increasing the blood flow above the resting level. Such changes for normal (AA) subjects are illustrated by the unshaded bars in Fig. 5. Three additional cases are represented in Fig. 5 to illustrate the effects of the

1.2

-

1.0

-

5 G

0.8

-

>

0.6

-

z I= g

0.4

-

0.2

-

1.8

-

CAPILLARY

p02

CAPILLARY

t

[02] 1

a 2 r 5 z k

CAPILLARY RESISTANCE

BLOOD

FLOW

t

l.B1.4-

0 z

1.2

;

l.O-

$

0.8

it

-

-

0.6

-

0.4

-

0.2

-

25%

125% % INCREASE

25% IN METABOLIC

DEMAND

FIG. 5. Effects of increasing tissue metabolic demand on capillary pOz , capillary [O,], capillary resistance, and blood flow. Unshaded bars: normal hematocrit, viscosity, and HbO, affinity; dotted bars: reduced hematocrit alone (22%); cross-hatched bars: reduced hematocrit and increased blood viscosity according to Eq. (6); solid black bars: reduced hematocrit, increased viscosity, and decreased Hb02 affinity (PsO = 37 mm Hg). Value of each variable in the resting condition for AA blood is used to normalize the results. Normal resting state values are identical to those in Fig. 4.

OXYGEN

TRANSPORT

IN SICKLE

CELL

DISEASE

203

major characteristics of SS blood on the response to increased metabolic demand, similar to those shown in Fig. 4 for resting condition. In anemia alone the low hematocrit (dotted bars) reduces the blood viscosity, thus causing a lower capillary resistance and a higher blood flow than in normal subjects; the capillary pOz and [O,] become lower due to the reduced oxygen carrying capacity. When the increased red cell rigidity of deoxygenated sickle cells is superimposed on the decreased hematocrit (cross-hatched bars), the increase in blood viscosity leads to an elevation of capillary resistance, which is slightly higher than that found in normal subjects, and a decrease in blood flow. As a result, additional oxygen extraction is required and capillary ~0, and [O,] are further reduced as compared to the case of anemia alone. When the increased PsOof sickle cell blood is also included along with reduced hematocrit and increased viscosity (solid black bars), there is a further reduction of capillary [O,], an increase in capillary resistance, and a decrease in blood flow; however, there is an improvement in capillary p02 due to the reduced HbOz affinity. The beneficial effect of the reduced HbOz affinity on capillary pOz diminishes as the metabolic demand increases from +25 to + 125%. Capillary recruitment occurs in response to increased metabolic demand. Considering the normal resting capillary density as 1.0, increasing the tissue metabolic demand induces increases of capillary density in the four conditions as follows. Increases in the metabolic demand by 25 and 125%, respectively, cause the capillary density to rise from 1.0 to 1.2 and 1.9 in normal subjects, from 1.3 to 1.6 and 2.6 in the presence of anemia, from 1.6 to 1.9 and 3.0 in anemia accompanied with increased blood viscosity, and from 1.3 to 1.6 and 2.8 in anemia with increased blood viscosity and reduced HbOz affinity. Effects of Decreased Arteriolar Blood Flow on Oxygen Delivery The occlusion of capillaries by sickle cells, through a reduction in capillary density, can cause an increase in peripheral resistance and a decrease in arteriolar blood flow. Arteriolar blood flow can also be reduced by other factors, e.g., a decrease in arterial pressure or an increase in arteriolar resistance. A decrease in incoming flow would reduce the oxygen supply and result in an increased oxygen extraction, a lower venous oxygen saturation and more sickling. The effects of decreasing blood flow to two levels (- 10 and -40%) have been simulated for the same four conditions described in the preceeding section (Fig. 6). In the case of normal blood, decreasing blood flow results in decreases in capillary pOz , [O,], and resistance; the decrease in resistance reflects an increase in capillary density due to precapillary dilation in response to tissue hypoxia, as illustrated by the unshaded bars in Fig. 6. The anemia in sickle cell disease (dotted bars) causes an enhancement in the magnitude of the decreases in PO,, [O,], and resistance. The combination of increased viscosity and reduced hematocrit (cross-hatched bars) leads to an increase in capillary resistance, and further lowers the capillary pOz and [O,]. The superposition of a decreased HbOz affinity (solid black bars) serves to elevate the capillary p02, even though the increase in capillary resistance and the decrease in [O,] are exacerbated. As the blood flow continues to decrease from- 10 to -4O%, however, the ability of the reduced HbOz affinity to maintain the critical tissue p02 becomes limited.

204

VAYO ET AL.

CAPILLARY 5

1.0

G ,

0.8

p F: g

a

[O,]

0.8 0.4

:

0.2

z8 8

1.2 1.0

5

0.8

g

0.6

L

0.4

RESISTANCE

0.2

-10%

-40% % DECREASE IN BLOOD FLOW

FIG. 6 Effects of decreasing arteriolar blood flow on capillary PO,, capillary [O,], and capillary resistance. Unshaded bars: normal hematocrit, viscosity, and HbOl affinity; dotted bars: reduced hematocrit alone (22%); cross-hatched bars: reduced hematocrit and increased blood viscosity according to Eq. (6); solid black bars: reduced hematocrit, increased viscosity, and decreased HbO, aftinity (P5,, = 37 mm Hg). Value of each variable in the resting condition for AA blood is used to normalize the results. Normal resting values are identical to those in Fig. 4.

In a manner similar to the example on the effects of increased metabolic demand, capillary recruitment is a compensatory response to decreasesin arteriolar blood flow. When the flow is reduced by 10 and 40%, respectively, the capillary density is 1.O and 1.3 for the normal case, 1.4 and 2.2 in the presence of anemia, 1.8 and at least 4.0 (the maximum capillary density permitted in the GrangerShepherd model) in anemia with increased blood viscosity, and 1.5 and 4.0 in anemia combined with increased blood viscosity and decreased HbOz affinity. DISCUSSION The model developed by Granger and Shepherd (1973) has been extended to analyze microvascular oxygen transport in sickle cell disease by incorporating the anemia, abnormal blood viscosity and reduced Hb02 affinity in this condition. The application of this model to simulate the effects of increased tissue metabolic demand and decreased arteriolar blood flow demonstrates that such modeling allows an assessment of the individual properties of sickle cell blood and an analysis of the factors responsible for the disturbances in microvascular flow and oxygen transport in this disease. There are other factors that contribute to the clinical presentation in sickle cell disease that have not been considered here. Among these are arterial desaturation, arteriovenous shunting, and the time required for sickling. The kinetics of the sickle hemoglobin gelation process may be of crucial importance in determining the degree of rheological abnormalities. Furthermore, the apparent viscosity of the blood used in this study was taken

OXYGEN

TRANSPORT

IN

SICKLE

CELL

DISEASE

205

from bulk viscometry rather than from measurements made in narrow channels similar to the size of capillaries. Using polycarbonate sieves with S-pm pores, Usami et al. (1975a) determined the dependence of the microsieving resistance of sickle cells on ~0~. The relative resistance from the microsieving data, which reflects primarily a pore entrance resistance, shows a stronger dependence on ~0, than that determined from the viscosity measurements, which reflect the rigidity of the SS cells in a bulk suspension. The dependence of sickle cell blood viscosity on the hematocrit and degree of saturation (Fig. 2) was derived from data determined at a shear rate of 208 see-‘. For lower shear rates, the sickle cell blood viscosity is higher than that shown in Fig. 2, which may significantly affect tissue oxygenation by further increasing resistance. Additional modifications incorporating these rheological characteristics of sickle cell blood may improve the quantitative aspects of this model, but would not alter the basic trends illustrated here. The major aim of therapy in sickle cell disease is to prevent the occurrence of the crisis episode. This can be accomplished by decreasing the residence time of the red cells in the capillaries and by inhibiting the hemoglobin gelation process. Recently, considerable effort has been devoted to the development of antisickling agents which would interfere with the gelation of sickle hemoglobin and thus improve capillary flow and tissue oxygenation. However, the effect of these drugs on Hb02 atIinity has not received appropriate attention. In moderate degrees of anemia (hematocrit > 25%), the decrease in HbOz affinity is sufficient to counteract the decrease in oxygen carrying capacity of the blood and maintain tissue oxygenation (Fig. 3). It is of interest to note that clinical symptoms are rarely manifested at moderately reduced hematocrit levels. It is only when the hematocrit is very low that the decrease in HbOz affinity is insufficient to maintain tissue oxygenation (Fig. 3) and that a marked increase in cardiac output is then required to provide adequate oxygen delivery (Sproule et al., 1960). Some of the antisickling drugs currently under investigation, such as phenylalanine and dimethyl adipimidate (DMA), cause an increase in Hb02 affinity. While the increased HbOz affinity tends to improve the venous oxygen saturation and thus decrease the number of sickled cells, it also negates an important compensatory mechanism to maintain tissue oxygenation by decreasing the amount of oxygen that can be unloaded to the tissue. These considerations indicate that the effect of antisickling drugs on HbOz affinity should be carefully examined. In conclusion, the major characteristics of sickle cell blood have been incorporated into the model of Granger and Shepherd (1973) in an effort to evaluate their individual contributions to oxygen delivery. Furthermore, the simulations indicate that, while the blood viscosity is a significant factor determining capillary flow, the decreased HbOz tin&y is important for maintaining proper tissue oxygenation and should be considered in investigations of pharmacologic therapies for sickle cell disease. The versatility of the original model has permitted the modifications described here and will allow the incorporation of additional parameters as more extensive research continues to elucidate the mechanism of sickle cell disease. ACKNOWLEDGMENT The authors are indebted to Dr. Harris J. Granger for providing a copy of the computer program of the original Granger-Shepherd model.

206

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CHIEN, S., USAMI, S., JAN, K.-M., SMITH,J. A., ANDBERTLES,J. F. (1976). Blood rheology in sickle cell disease. In “Molecular and Cellular Aspects of Sickle Cell Disease” J. I. Hercules, G. L. Cottam, M. R. Waterman, and A. N. Schechter, eds. DHEW No. 76-1007, pp. 277-303. NIH, Bethesda, Md. DE FURIA, F. G., MILLER, D. R. AND CANALE, V. C. (1974). Red blood cell metabolism and function in transfused beta-thalassemia. Ann. N. Y. Acad. Sci. 232, 323-332. FESTER, R. S., AND ASAKURA, T. (1979). Oxygen dissociation curves in children with anemia and malignant disease. Amer. J. Hematol. 7, 233-244. GRANGER, H. J., AND SHEPHERD,A. P. (1973). Intrinsic microvascular control of tissue oxygen delivery. Microvasc. Res. 5, 49-72. HORNE, M., III. (1981). Sickle cell anemia as a rheologic disease. Amer. J. Med. 70, 288-298. MAN~REDI, F., SPOTO, A., SALTZMAN, H., AND SIEKER, H. (1960). Studies of peripheral circulation during sickle cell crisis. Circulation 22, 602-607. MILLER, D., WINSLOW, R., KLEIN, H., WILSON, K., BROWN, F., AND STATHAM, N. (1980). Improved exercise performance after exchange transfusion in subjects with sickle cell anemia. Blood 56,

1127-1131. MILNER, P. (1974). Oxygen transport in sickle cell anemia. Arch. Intern. Med. 133, 565-572. MILNER, P. (1981). The clinical effects of Hb S. An overview. In “The Function of Red Blood Cells:

Erythrocyte Pathobiology” Alan R. Liss, Inc. New York, NY, pp. 297-320. ODUNTAN, S. (1969). Blood gas studies in some abnormal haemoglobin syndromes. &it.

J. Haematol.

17, 535-541. SPROULE, B., MITCHELL, J., AND MILLER, W. (1960). Cardiopulmonary physiological responses to heavy exercise in patients with anemia. J. C/in. Invest. 39, 378-388. USAMI, S., CHIEN, S., AND BERTLES,J. (1975a). Deformability of sickle cells as studied by microsieving. J. Lab. Clin. Med. 86, 274-282. USAMI, S., CHIEN, S., SCHOLTZ, P., AND BERTLES,J. (1975b). Effect of deoxygenation of blood rheology in sickle cell disease. Microvasc. Res. 9. 324-334.