Diffusion coefficients of oxygen and hemoglobin as obtained
simultaneously from photometric determination of the
oxygenation of layers of hemoglobin solutions
Citation for published version (APA):
Kreuzer, F. J. A., Spaan, J. A. E., & Wely, van, F. K. (1980). Diffusion coefficients of oxygen and hemoglobin as
obtained simultaneously from photometric determination of the oxygenation of layers of hemoglobin solutions.
Pflügers Archiv : European Journal of Physiology, 384(3), 241-251. https://doi.org/10.1007/BF00584558
DOI:
10.1007/BF00584558
Document status and date:
Published: 01/01/1980
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Pflfigers Arch. 384, 241-251 (1980)
Pflfigers Archiv
European
Journal
of Physiology
9 by Springer-Verlag 1980
Diffusion Coefficients of Oxygen and Hemoglobin
as Obtained Simultaneously from Photometric Determination
of the Oxygenation of Layers of Hemoglobin Solutions*
J. A. E. Spaan 1, F. Kreuzer 2, and F. K. van Wely
i
Department of Physiology and Physiological Physics, Leiden University Medical Center, Wassenaarseweg 62, Leiden 2 Department of Physiology, University of Nijmegen, Geert Grooteplein Noord 21a, Nijmegen, The Netherlands
Abstract. The oxygenation of layers of deoxygenated
hemoglobin solutions after a sudden exposure to a gas
containing oxygen at a partial pressure PI has been
studied by a photometric method. Layer thicknesses
varied between 50 and 250 ~tm, hemoglobin concen-
trations between 0.1 and 0.34 kg/1, and oxygen partial
pressures between 4.65 and 93.1 kPa (35 and
700mmHg). The diffusion chamber containing the
layer of hemoglobin solution permitted a step change in
gas atmosphere without changing the optical apparatus
constant.
The following results were obtained:
1. The oxygen saturation increase is independent of
the layer thickness when expressed as a function of time
divided by layer thickness squared (normalized oxy-
genation time). This justifies the assumption of chemical
equilibrium between oxygen and hemoglobin in the
range considered.
2. The oxygen saturation increases proportionally
to the square root of time over a wide range of
oxygenation as expected. This range reaches to almost
100% oxygenation at P1 = 93.1 kPa (700mmHg) but
less far as P1 is lower. Thus at high P1 values there is a
sharp boundary between the oxygenated and deoxy-
genated part of the layer allowing the application of the
advancing front concept.
3. Fitting the theoretical equations derived in a
preceding paper to the experimental results provides
simultaneous values of the oxygen permeability (or,
knowing oxygen solubility, of the oxygen diffusion
coefficient) and of the hemoglobin diffusion coefficient.
These values agree fairly well with values obtained by
other authors from experiments yielding the diffusion
coefficients of oxygen or hemoglobin separately.
* The experiments were performed in the Biomedical Engineering Section of the Department of Production Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands
Key words: Oxygenation of hemoglobin - Photometry
- Facilitated diffusion - Diffusion coefficient of
oxygen - Diffusion coefficient of hemoglobin.
Introduction
In a previous paper we analyzed the nonsteady-state
transfer of oxygen in layers of hemoglobin solutions
based on the application of diffusion equations to the
diffusion of both oxygen and hemoglobin assuming
chemical equilibrium between oxygen and hemoglobin.
This analysis made it possible to derive values of the
oxygen permeability (or the oxygen diffusion coef-
ficient when knowing oxygen solubility) and the dif-
fusion coefficient of hemoglobin simultaneously from
measuring the oxygenation time.
The present paper describes the experimental pro-
cedure of measuring the oxygenation time in layers of
hemoglobin solutions 50-250 ~tm thick in the
presence of oxygen driving pressures of 4.65 - 93.1 kPa
(35 - 700 mm Hg).
Numerical values of the diffusion coefficients of
oxygen and hemoglobin were obtained from the oxy-
genation experiments over a wide range of hemoglobin
concentrations and compared with data from the
literature.
The results with 93.1 kPa oxygen driving pressure
were compared with those of Klug et al. (1956) after
correction of their data.
Methods
Principle of Photometric Method
A layer of deoxygenated hemoglobin solution 5 0 - 250 gm thick was spread on a leveled glass plate. This layer was abruptly exposed to a
242 Pfltigers Arch. 384 (1980) gaseous atmosphere o f a known oxygen partial pressure PI as the
driving pressure for oxygen transfer into the layer.
The oxygen saturation o f the layer increased with time up to its maximally possible value S 1 depending on Px as determined by the oxygen saturation curve o f the solution.
As shown in our preceding theoretical paper (Spaan et al., 1980) the initial phase of the oxygenation process can be described by :
/ >
(1)
V~ = = X/ q~ d2
where
s-s~
dimensionless oxygen saturation,
SI-&
S = oxygen saturation (fractional),
St = initial oxygen saturation (fractional),
$1 = final oxygen saturation (fractional),
t = time (s),
t~ = oxygenation time (s),
t/d 2 = normalized time (ms/gin2),
tl/d 2 = normalized oxygenation time (ms/~tm2).
The oxygenation time is the time needed to obtain complete oxygenation ifeq. (1) would apply throughout the whole oxygenation process. The normalized time has been introduced since then the normalized oxygenation time is independent o f layer thickness. The present experimental method is based on the determination o f q / d 2 from measured relationships between ~ and ]/7.
This process o f oxygenation was measured by recording the change in absorption of light at a wavelength o f 670 n m (band width 40 nm) from a parallel beam o f light transmitted perpendicularly through the layer.
Due to the specific nature o f our problem, i.e., 1. the hemoglobin solution not being contained in a cuvette of primarily known layer thickness, 2. the possible presence of relatively high concentrations of inactive hemoglobin, and 3. the use of g/1 as unit o f hemoglobin concentration rather than mmol, our application of Lambert- Beer's law differs from present convention in spectrophotometry particularly in clinical chemistry. Therefore we will first define our use of terms and dimensions. Lambert-Beer's law is: log I/I o = - e c d = - e d , where:
I = intensity of light after passing a layer of solution with
concentration c,
I 0 = intensity of light after passing the same layer where c = 0 (blank),
e = extinction coefficient of light-absorbing species per gram
(1 " g - a . m m - ~ ) ,
c = concentration of light-absorbing species (g/l), d = layer thickness (mm),
e = e e = absorption coefficient (mm-*).
So according to the law of Lambert-Beer the intensity of the light beam as detected by the light sensor is (see also Spaan et al., 1977):
log I(t) = log I~-{eo[1 - S(t)] + el. ,S'(t)} d (2)
where l(t)
#(t)
eo el. d log I a= intensity o f the light after passing the layer, = average oxygen saturation of the layer,
= absorption coefficient of the solution at zero oxygen saturation ( m m - 1),
= absorption coefficient of the solution at full oxygen saturation (ram- 1),
= layer thickness, = apparatus constant.
The change in dimensionless oxygen saturation now can be obtained from the change in measured light intensity from:
log [I(t)/lo]
( t ) - - - ( 3 )
1og(11/lo)
where
Io = intensity of light recorded at S = S~, /1 = intensity of light recorded at S = S 1. The layer thickness can be determined from:
log (I1/Io)
d - . (4)
(e0 -- et) (S, - Si)
Obviously eqs. (3) and (4) hold only when no geometrical changes occur in the hemoglobin layer and other media through which the light is transmitted.
Since the oxygen saturation of the layer initially changes proportionally to the square root of time, the experimentally obtained relation between light intensity and time was fitted by the following equation:
log I ( t ) = x a ] / t + x 2 (5)
Io
where x~ and x 2 are constants. Theoretically the constant x 2 has to be zero, but it has been added since exceptional experiments showed a finite value of x 2 probably due to a small change in apparatus constant (log In) when changing the gas atmosphere.
Combination of eqs. (1), (3), (4) and (5) results in the following equation for the normalized oxygenation time:
,1 =l (eo-el.Ul- S,)t
d2 ( xl j . (6)
Note that for the determination o f h / d z initial (lo) and final (11) light intensity as well as the actual layer thickness are not needed.
Determination o f Oxygen Permeability and Diffusion Coefficient o f Hemoglobin from Oxygenation Experiments at Varying Oxygen Driving Pressures
In the preceding theoretical paper we derived a relationship (eq. 16) defining the normalized oxygenation time tl/d 2 as a function o f 1/1'1 in terms of the product of several factors. In particular we established a functionf~ depending predominantly on P50 of the saturation curve and l/P1, and fz being the nonsteady-state facilitation factor pre- dominantly depending on D~ and lIP> Whereas for Ps0 ~ 0 or P1 --. oe (rectangular saturation curve) fl = 1 by definition, f1 is 1 - 1 . 1 for our present saturation curve characterized by Ps0 = 1.6kPa (12 mm Hg), a n d / ' 1 between 4.65 and 93.1 kPa (35 and 700 mmHg). For On = 0 (or PI ~ oo), f2 is maximum and equals 1; thus f2, the nonsteady-state facilitation factor, manifests itself by a reduction of the oxygenation time due to the diffusion o f hemoglobin.
In absolute quantities the theoretical model resulted in the following equation when assuming S i = 0 and $1 = 1 (eq. 16 o f preceding paper): q h 1 1 where h Z - ~D c
h = oxygen binding capacity of the hemoglobin solution (mol/1),
c~D e = oxygen permeability o f the solution (mol. s - 1 . m - 1 . Pa 1),
c~ = oxygen solubility of the hemoglobin solution (mol . 1-
P a - 1),
D c = diffusion coefficient o f oxygen (ms/~tm2),
/)1t = diffusion coefficient o f hemoglobin (ms/gm2), Pl. = oxygen driving pressure (kPa).
J. A. E. Spaan et al. : Oxygenation o f Layers o f Hemoglobin Solution 243 / / / / /
/,d,
"./d
=/.4,
/.d,
/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / //././///\\\\\\\\\\\\\~XI]].
b ",,\\\
a b\
Fig. 1 a and b. Outline of the experimental apparatus, a shows a cross-section and b presents a view from the top. I diffusion chamber;//cylindrical housing of the diffusion c h a m b e r ; / / I s t e p - s h a p e d glass plate to carry the hemoglobin layer; I V t o p plate mounted rigidly on the b o t t o m of the housing; V double-glass upper window; VI teflon turntable provided with two chambers; VII nitrogen gas chamber; VIII oxygen gas chamber; 1Xpivoted arm on which both the optical fiber and the photocell are mounted; Xinlet and outlet channels for the gases; X I arm for turning the teflon table; X I I end of the fiber mounted in a ball joint, allowing to adjust the direction of the light beam; X / / I photocell
The theoretical diffusion model expressed by this equation was fitted to the experimental data of the normalized oxygenation times plotted against the reciprocal oxygen driving pressure by the method of least squares, applying the polygonal approximation outlined in the preceding paper and varying the parameters Z and Dn. According to eq. (6) the experimental normalized oxygenation time depends on the value of (e 0 - el) 2 with (S~ - S~) = 1. Therefore according to eq. (7) the value of Z also depends on ( e o - e l ) 2. Moreover, when the permeability c~D c is determined from the estimated value of Z, the oxygen binding capacity o f the solution has to be known. However, the estimated value of DH is independent of any other quantity and follows directly from the fitting procedure.
Apparatus and Experimental Procedure
Figure ] a shows a cross-section of the diffusion chamber as well as its housing and the essential parts of the optical system. Figure l b presents a view from the top.
The diffusion chamber (I) is housed in a metal cylinder (II) of 14.5 cm diameter suspended in a thermostated water bath of 25~ and basically consists of three parts:
1. a profiled glass plate (III) carrying the hemoglobin sample and being mounted at the bottom of the cylinder;
2. a top plate (IV) containing a double glass window (V) designed to be easily mountable after a hemoglobin layer has been spread;
3. a turntable (VI) made of teflon between the bottom of the cylinder and the top plate. This table is provided with two holes to act as gas chambers for nitrogen (VII) and oxygen (VIII) respectively. By appropriate design and careful matching, leakage between gas spaces and outer atmosphere can be minimized. Inlet and outlet ports (X) permit continuous flushing with gas. By turning the table the hemoglobin film can be exposed either to nitrogen or to oxygen. The change from nitrogen to oxygen can thus be achieved virtually as a step function.
The optical system consisting of an optical fiber and a sensing diode is mounted on a pivoted arm (IX) which permits either to spread the hemoglobin layer or to clean the glass plate. Moreover, an angle-measuring device on the pivoted arm makes it possible to check the degree of uniformity of the layer thickness. This design provides an identical optical apparatus constant for both the "nitrogen" and "oxygen" situation of the device.
The gas mixtures were prepared from gas tanks, containing 5 CO2 and 95 ~ nitrogen or 95 ~o oxygen or air, by means o f a gas mixing pump (W6sthoff). The accuracy o f the mixing was within 0.5 ~ as tested with a gas chromatograph. The gases were humidified by means of washing flasks containing thermostated water. The time for changing the Po, within the oxygen chamber from 93.1 to 0 k P a (700 to 0 m m H g ) by readjusting the setting of the pump was 10 min. First the diffusion chamber was exposed to nitrogen. By means of a micro syringe (Hamilton) and a glass spatula a small amount of completely deoxygenated hemoglobin solution (up to 50 gl depending on the desired layer thickness) was spread on the glass plate after removing the glass window. After positioning the layer the window was reinstalled. Next the pivoted arm was adjusted to direct the light beam to the center of the layer of hemoglobin solution.
Subsequently the value of I o was recorded. After a certain lapse of time the teflon turntable was moved to expose the hemoglobin layer to the oxygen-containing gas, whereafter the oxygenation process started. The course of oxygenation was recorded by the change in the light absorption by hemoglobin. After completion of the oxygenation the value of 11 was recorded. In case the oxygen partial pressure of the gas was chosen to be $1 < 1, the Po, was increased to determine S 1.
A thermostated (_+ 0.1 ~ C) light-emitting diode (LED) was used as a light source. The peak intensity of the L E D (Monsanto MV/10B) was at 670 nm, the width of the distribution at 50 ~o peak intensity was 40 nm. The light intensity of the L E D alternated harmonically at a
244 Pflfigers Arch. 384 (1980) frequency of 1,552 Hz. The light was conducted by fiber optics to pass
through the hemoglobin layer, and some of it was received by a light- sensing diode (Monsanto M D 2). The output of the preamplifier of the oximeter consisted of the superposition of a DC voltage and an A C voltage. The DC component was proportional to the intensity of the light received, the AC component was filtered out by a low-pass filter with a cut-off frequency o f 20 Hz.
A digital and analog recording system have been used simul- taneously. The digital recording system used a HP ten-channel scanner. The scanner sequentially connected the output o f the oximeter and the output of an analog time clock to an analog-digital converter. The digital numbers were punched in paper tape. The information on paper tape was used for computer calculations.
The analog time clock consisted o f a circuit integrating a constant voltage. The integrator was started by a sharp rise in the output of the oximeter. This corresponded to the moment when the oxygen chamber in the teflon turntable was turned over the hemoglobin layer, thus starting the oxygenation process. A n electrical circuit to trigger the data-logging has been designed in order to make the interval time A t between two measuring cycles proportional to the square root o f time. As far as the oxygen saturation also changed with the square root of time, measurements were obtained at points of equal increase in saturation.
A n analog recorder (Houston 2000) was used for direct obser- vations during the experiments. From the output signal of the oximeter connected to the recorder a constant voltage was sub- tracted; thus small changes in intensity could be recorded on full scale.
Fresh packed red blood cells (human blood group 0 +) were washed three times with saline (9~o NaC1) and separated by centrifuging (6,000 g, 10 rain). The cells were lyzed by adding 60 ml of distilled water to 100ml of packed cells. The red cell stroma was removed by adding 40 ml of toluene to the solution. After shaking the mixture the toluene was separated from the hemoglobin solution by centrifuging (6,000 g, 20 min). The hemoglobin solution was sucked off from underneath the toluene (hemoglobin concentration between 0.16 and 0.19kg/1) and poured into specially designed tonometer bottles. The hemoglobin solutions were diluted by adding saline. High hemoglobin concentrations were obtained by ultra-filtration (Diaflow, Amicon Co.).
The tonometer bottles were turning continuously for a couple of hours while being flushed with a humidified gas mixture o f 95 ~ N 2 and 5 ~ CO2. The O2 content of the solutions was checked with a Lex-O2-Con. As soon as the oxygen concentration became unde- tectably small (less than 10-4mol/1) the bottles were pressurized (120 kPa), closed and stored at 4~
Hemoglobin concentration and percentage of methemoglobin were determined by a single-beam spectrophotometer using the cyanide method of Zijlstra and Van Kampen and the method according to Evelyn and Malloy respectively, as described by Van Assendelft (1970). By following this procedure of preparation and storage the methemoglobin content usually proved to be less than 1 of the total hemoglobin concentration.
The oxygen binding capacity was analyzed on five independently prepared hemoglobin solutions, using the Lex-O2-Con for the determination o f the total oxygen concentration from which the physically dissolved oxygen was subtracted, and a spectrophotometer for the determination of the total hemoglobin concentration and o f the fraction of methemoglobin. In all these cases the oxygen binding capacity was found to be 1 - 4 ~ less than the value calculated from the hemoglobin concentration (corrected for the fraction o f methe- moglobin present) as measured by the spectrophotometer. Recently very accurate measurements o f the oxygen binding capacity were performed by Dijkhuizen et al. (1977) who also found an oxygen binding capacity lower than predicted from the concentration of the supposedly active hemoglobin (6.2 - 10 5 moi/g Hb). We adopted a
1 , " ,
S .8
#
~r''''J" D'----~
.6
j~
.4
x r
/[/
r
~
i
;
;
Poz (kPa)
Fig. 2. Experimental data for determining the value of Pso of the actual saturation curve holding in the present experiments. The solid line is the Standard Dissociation Curve (SDC) with the Po~ axis attenuated by a factor o f 2.21 in order to fit the experimental points. It is concluded that the actual P50 is 1.6 kPa (12 mm Hg = 26.6/ 2.21 mm Hg). The broken line intercepts the line S = 1 at Po~ = 3.5 kPa (26.2 mm Hg). The polygonal approximation used in the parameter estimation program is defined by the broken line for Po2 < 3.5 kPa, and S = 1 for Po2 > 3.5 kPa
binding capacity of 6.07.10 5 (1 - Sm)mol O2/g Hb, where Sm is the fraction of methemoglobin present.
The difference in the light absorption coefficients e 0 - el for each hemoglobin solution was corrected for methemoglobin by:
e0 - el = (eHb - - eHbO2) (1 - - SHi ) b (8)
where
8Hb--eHbO2 = difference in the extinction coefficients ofhemoglobin and oxyhemoglobin for the light used (1 9 g 1. m 1),
Si4i = percentage of methemoglobin,
b = total hemoglobin concentration (g/l).
Clearly eq. (8) does not correct for inactive hemoglobin other than methemoglobin.
For three independently prepared fresh hemoglobin solutions o f
171, 185 and 290g/1 hemoglobin concentration and Sin< 0 . 5 ~
values of 4.33, 4.32 and 4.381. m - 1. g 1 respectively where found for eHb using the wedge method o f Spaan et al. (1977); for enbo_~ a value o f 0.40 1 - m - 1. g - 1 was found. Subtracting this value of 0.40 from the mean of 4.34 for ~HU provided a @ H b - - S H b O 2 ) of 3.94 1 9 m -1 9 g-1 which was used in all experiments.
A saturation curve was measured over its entire range for two different hemoglobin solutions. A Radiometer electrode was used for the Po2 determination and a Lex-O2-Con for the estimation of oxygen saturation. Three times a few points of the saturation curve were obtained using the diffusing apparatus. For this a layer of hemo- globin solution was equilibrated with gases of different but known Po~. The saturation was measured optically. The solid line connecting the data points in Fig. 2 shows the saturation curve thus obtained. Its Ps0 is 1.6 kPa (12 m m Hg) as compared with a Ps0 of 3.55 kPa (26.6 m m Hg) for the standard dissociation curve (ratio = 26.6/t2 = 2.21). The polygonal approximation o f this saturation curve (see previous paper) is indicated by the broken line.
Evaluation of Experimental Method and Procedure
Optical System. The bandwidth o f the L E D is in the order of 40 nm, which is rather considerable when comparing it to the part of the absorption spectrum relevant for the measurements. Within this part
J. A. E. Spaan et al. : Oxygenation of Layers of Hemoglobin Solution 245 of the spectrum the extinction coefficient of methemoglobin changes
by a factor of 8, that of deoxyhemoglobin by a factor of 2, and that of oxyhemoglobin by a factor of 0.7. However, it can be shown (see Appendix) that the law of Lambert-Beer still may be applied when introducing an adapted extinction coefficient (8*) which may differ essentially from the extinction coefficient at the peak wavelength of the emission spectrum of the LED. It may easily be seen that if different light-absorbing species are present in the solution the summation rule of absorbing species, thus in fact eq. (2), remains valid when using the adapted extinction coefficient.
Hemoglobin Layer.
The geometry of the hemoglobin layer, at least at the point of incidence of the light beam, has to remain unchanged throughout the course of oxygenation. Apart from qualitative visual observation the behavior of the hemoglobin layer with time can only be judged quantitatively by means of light absorption recordings. Since the light absorption at S = 0 (zero saturation) is ten times larger than that at S = 1 (full saturation), the stability of the hemoglobin layer has to be checked while the diffusion chamber is flushed with nitrogen.Several factors where recognized as sources of experimental error:
1. the hemoglobin layer has not been spread anaerobically, 2. the gas flow is not sufficient to compensate for air leakage through the sealing into the chambers,
3. the glass plate has not been leveled sufficiently resulting in a convective flux due to gravity,
4. the gas flow is too high and disturbs the layer surface, 5. the layer is drying.
Appropriate gas flow settings were found to be 2 ml/s N 2 before starting the oxygenation and 0.4 ml/s O2 during oxygenation. First a sample holder having a uniform cylindrical well was used where however the layer surface was slightly bent upward. Although this bend need not impede the measurements as long as the light beam traverses an only small area in the center, the stability of the layer is very sensitive to deviations from horizontal. A significant improve- ment was achieved by applying a sample holder with a step-shaped well (steps of 50 Ixm each) so that the hemoglobin layer now adhered to the well edges. With this sample holder a tilting angle of up to 0.7 degree did not affect the stability of the signal.
With all these precautions concerning the hemoglobin layer and gas flow the measuring signals before and after oxygenation remained stable for a time longer than needed for even the longest oxygenation studied (30 min). Drying of the layer was not observed. Moreover the experimental results proved to be independent of the time (up to 30 min) the layer had been exposed to the humidified nitrogen flow.
Range of Applicability
The change in light absorption of a hemoglobin layer due to oxygenation depends on both the hemoglobin concentration (b) and the layer thickness (d). The oximeter is sensitive enough to detect a change of 10 -3 in light absorption. Eq. (2) relates the absorption error A S in the measurement of oxygen saturation to both the error in light absorption measurement and the product (e0-el)b d [= ( e o -
el)d ].
Using a value of 4 l 9 m- ~ . g i for (8o-E1) one finds that the lower limit of the productbd
is (10Z/AS)
m ' g 9 1 ~ whereAS
isgiven in saturation percent.
With the present sample holder allowing a maximal layer thickness (d) of 250 gm and the required accuracy of 1% saturation
(AS)
the hemoglobin concentration should not be lower than 0.08 kg/l.Solubility of Oxygen in Hemoglobin Solutions
In order to calculate the diffusion coefficient of oxygen from the oxygen permeability the oxygen solubility needs to be known.
Sendroy et al. (1934) showed that the solubility of oxygen in water (%2o) is decreased by the presence of dissolved salts but increased by dissolved hemoglobin. Sendroy et al. (1934) are the only authors reporting oxygen solubility of hemoglobin solutions (C~Hb). They found no difference in solubility increase per g Hb between stroma- free hemoglobin solutions and suspensions of red cells in saline at 38~ Hedley-Whyte and Laver (1964) showed that the ratio
C~B/C~H2
o
(c~ B = oxygen solubility in blood) is independent of temperature. Although this ratio does not linearly depend on hemoglobin con- centration, the results of Christoforides and Hedley-Whyte (1969) may be approximated to within 1.5 % by:
ae/C~H2 o -- 0.9 + 0.000312 [Hb] (9)
where [Hb] is in g/1.
The oxygen solubility of the hemoglobin solutions was calculated from aH2o from Hodgman et al. (1957; p. 1606) [%2o = 1.25 . i 0 - 8 tool. 1 1. Pa- 1 (0.0283 ml
Oz/cc/atm)
at 25 ~ C], taking into ac- count the decrease of solubility due to dissolved salts according to Sendroy et al. (1934) (Ae = 3.2 " 1 0 - 9 mol 9 Pa 1 . mol 1 NaC1,[NaCI] approximately 0.07 mol/1) and the increase of solubility due to hemoglobin from eq. (9) leading to:
~Hb = ~XH20 (l ~- 0.000312 [Hb])- Ac~[NaC1]. (10)
Results
S e v e r a l series o f o x y g e n a t i o n e x p e r i m e n t s o n h e m o - g l o b i n s o l u t i o n s w i t h c o n c e n t r a t i o n s in t h e r a n g e o f 0.1 - 0.34 kg/1 w e r e p e r f o r m e d . T h e r e s u l t s o f o n e series o f e x p e r i m e n t s ( r e p r e s e n t a t i v e f o r a t o t a l o f five series) u s i n g a h e m o g l o b i n s o l u t i o n o f 0.29 kg/1 a r e s h o w n in F i g . 3. T h e a p p l i e d o x y g e n d r i v i n g p r e s s u r e s P1 w e r e in t h e r a n g e o f 4 . 6 5 - 9 3 . 1 k P a ( 3 5 - 7 0 0 m m H g ) . T h e m e a s u r e d o x y g e n a t i o n i n c r e a s e is p l o t t e d as a f u n c t i o n o f t h e s q u a r e r o o t o f t h e n o r m a l i z e d t i m e ( =t/d2).
T h e l a y e r t h i c k n e s s e s w e r e c a l c u l a t e d o p t i c a l l y b y eq. (4). T h e p o i n t s f o r d i f f e r e n t e x p e r i m e n t s at e q u a l v a l u e s o f PI a r e c h a r a c t e r i z e d b y d i f f e r e n t s y m b o l s . F r o m e a c h e x p e r i m e n t o n l y p a r t o f t h e d a t a p o i n t s a r e s h o w n t o a v o i d c r o w d i n g o f t h e f i g u r e . T h e s t r a i g h t lines in Fig. 3 w e r e o b t a i n e d b y l i n e a r r e g r e s s i o n a n a l y s i s ( a p p l y i n g eq. 5) u s i n g t h e d a t a p o i n t s f r o m a n y o n e o x y g e n a t i o n e x p e r i m e n t in t h e s a t u r a t i o n r a n g e w h e r e , as j u d g e d b y eye, t h e r e l a t i o n s h i p b e t w e e n 7 / a n d]/~/d 2
w a s a b o u t l i n e a r . T h e g e n e r a l c o u r s e o f t h e c u r v e s r e s e m b l e s t h a t d e r i v e d t h e o r e t i c a l l y as s h o w n in F i g . 1 o f t h e p r e v i o u s p a p e r . T h e v a l u e s o b t a i n e d f o r I Xzl in t h e s e e x p e r i m e n t s w e r e a l w a y s less t h a n 0.01 e x c e p t in 3 e x p e r i m e n t s . L i n e a r v a r i a n c e a n a l y s i s r e s u l t e d in a 95 % c o n f i d e n c e l i m i t o f x t o f less t h a n 0.01 x~. T h e r e p r o d u c i b i l i t y o f t h e e s t i m a t e d n o r m a l i z e d o x y g e n a t i o n t i m e s is b e t t e r t h a n 4 %. T h e d i f f e r e n c e b e t w e e n t h e c u r v e s m e a s u r e d at P1 = 2 7 . 9 k P a (210 m m H g ) s h o w n in Fig. 3 is d u e t o o x y g e n l e a k a g e i n t o t h e s a m p l e s y r i n g e , as c o u l d b e s h o w n b y a L e x - O 2- C o n d e t e r m i n a t i o n . O u t o f t h e e x p e r i m e n t s o f h i g h a c c u r a c y (95 c o n f i d e n c e l i m i t o f x ~ less t h a n 0.01 xa, s t e p - s h a p e d w e l l246 Pfliigers Arch. 384 (1980)
u;
. 8.6
. 4 . 2 0 ~ / / I I I 0 5 0 1 0 0 1 5 0 .,,(. 9 9 * 9 * / # PI d c u r v e k P a p m I 9 9 3 . 1 1 2 3 9 1 1 7:.
5 0 2 9 5 5 . 9 181 9 1 2 9 3 9 2 7 . 9 8 7 9 7 4 4 9 9 . 3 1 2 4 , 1 2 5 5 9 4.6 1 1 9913
2 0 O #t ~
(s~.m m ')
Fig.3. Computer plot o f the experimentally determined course o f oxygenation T as a function o f the square root o f normalized time
t/d 2.
[Hb]= 0.29 kg/1, (e o - el) = 1.14 r a m - ~./)1 varies between 93.1 and 4.65 kPa (700 and 35 mmHg), The straight lines in the figure were obtained by linear regression analysis on the linear part of the experimental curves. The layer thicknesses and the P~ values applied in each individual experiment are given in the table of the figure
sample holder, class A) and those of lower accuracy
(95 ~ confidence limit of xl less than 0.04 xl, cylindrical
well, class B) three complete series of class A experi-
ments are shown in Fig. 4. There is a curved re-
lationship between
t l / d 2and
1/P 1indicating
hemoglobin-facilitated oxygen transfer (see Fig. 6 in
preceding paper). There is fair agreement between the
experimental points and the solid lines obtained by
fitting the diffusion model to the data (eq. 7).
The values of the oxygen permeability
~Dcand the
diffusion coefficient of hemoglobin
DHare plotted
against hemoglobin concentration in Figs. 5 and 6
respectively. The dots refer to the class A experiments,
the crosses to the class B experiments. Linear variance
analysis resulted in 95 ~ confidence limits of 3 ~o for
~D cand 7 ~ for
D Hin the class A experiments; these
values are 8 ~ for
e D cand 15 ~ for
D Hin the class B
experiments.
In Fig. 5 data of
eD cobtained by Stroeve (1973;
Stroeve et al., 1976) are indicated by squares. His values
were determined from steady-state oxygen fluxes
through layers ofmethemoglobin solutions. The agree-
ment between his and our results, obtained by two
different techniques, is quite satisfactory. Our data of
the diffusion coefficient of hemoglobin (dots and
crosses in Fig. 6) appear to be a little higher than but
still close to those of other authors. However, since
corrections have been applied to the data of the other
authors, Fig. 6 will be discussed in detail below.
To assess the effect of an uncertainty in the oxygen
saturation curve on these values, a parameter esti-
mation was performed on two experimental series using
polygonal approximations of the saturation curves
with different values of Pso (0.06, 1.57, and 4.82 kPa).
The results are shown in Table 1. The variation is in
the order of 5 ~ for the oxygen permeability and in the
order of 30 ~ for the diffusion coefficient of hemo-
globin when
Psois changed from 1.57 to 4.82 kPa. The
actual Ps0 is known within 10 ~ (see Fig. 2). Assuming
a linear relationship between the variation in/'50 and
J. A. E. Spaan et al. : Oxygenation of Layers of Hemoglobin Solution 247 2 0 18 1 6 i = , 1 7 0 - - curve [Hb]
Pg/l
/
1 0,290 2 0.224 3 0.152 -74 1 4 .. / 7
98&
120
"E 8 6/
123/,.179/ :
1~ 4 ."21s "~51/112./,
laO / "
166
4 / 78,,, l~u // /
98 _,,~2"282//,1~o1~./"
2/ / / ' 1 6 2
k P a(
Pl
~ I 0 o 3 0 2 0 O0 2'0 4'0 6 01/Pi ) I1/M Pa)
Fig. 4. Experimentally determined normalized oxygenation time t / d 2 versus reciprocal oxygen driving pressure (l/P1). The results of three class A experimental series were obtained with hemoglobin con- centrations of 0.152, 0.224 a n d 0.29kg/1 (15.2, 22.4, and 29.0g%). The n u m b e r next to each point represents the respective layer thickness in ~tm. The curves were obtained by fitting the polygonal model to the experimental data on the basis of the m e t h o d of least squares by varying the parameters Z ( = h/o~Dc) and D m The solid curves are the curves of best fit
the values of
c~D cand DL,, it follows that uncertainties
in
c~D cand D H due to the uncertainty in Ps0 are in the
order of 0.12 % and 1% respectively.
A saturation curve with Ps0 = 0.06kPa is almost
rectangular and therefore also Hill's (1928/1929) mov-
ing boundary model could be applied. Our experiments
using a /'5o = 0.06kPa or Hill's model would have
resulted in an overestimation of
c~D cin the order of 2 %
and an underestimation of DH in the order of 12 % as
compared with an actual Pso of 1.57 kPa, which again
underlines the minor influence of the saturation curve
on the oxygenation process.
Eventually we compared our respective data with
those obtained by Klug et al. (1956). The values found
for the time needed to reach half saturation (q/2) at/'1
= 93.1 kPa (700 mm Hg) for a wide range of hemo-
globin concentrations are compared with their values in
Fig. 7. Since some correction had to be applied to the
data of Klug et al. Figure 7 will be considered further in
the Discussion.
_ - 3 ~ I
'm O. 'r20
o 15,~ lO
, I0
.1
~ x x i 9 i [] Stroeve 9 class A • class B B ~ x x [] I , I.2
.3
[ H b ] ( k g / l lf
Fig. 5. Values f o u n d for the permeability of oxygen c~D c as a function
of hemoglobin concentration (0.I kg/1 = 10g %) at T = 25~ The
dots correspond to the class A experiments, the crosses to the class B experiments, a n d the open squares to the experimental data of Stroeve (1973). The class A experiments have an accuracy of 3 %, the class B experiments of 7 %. The solid curve is adapted to the dots and crosses
.005
\
' ~ '
0 .1 . .3 .4
[Hbl ( k g / I )
Fig. 6. Values found for the diffusion coefficient o f hemoglobin D~q as a function of hemoglobin concentration (0.1kg/1 = I 0 g % ) at T
= 25~ The dots correspond to the class A experiments, the crosses
to the class B experiments. Open circles are data of Moll (1966) corrected for temperature and filter tortuosity. The solid line represents the data of Riveros-Moreno and Wittenberg (1972)
corrected for temperature. The broken line ( - - - ) represents the
data of Keller et al. (1971). The lower broken-dotted line ( . . . ) is the compromise curve of Kreuzer (1970) for uncorrected data ( T = 2 0 - 2 5 ~ found in the literature. The upper broken-dotted line ( . . . ) is adapted to the present results
D i s c u s s i o n
In the preceding theoretical paper the oxygenation
process has been approached by two models, a finite
and semi-infinite model, assuming chemical equilib-
rium between oxygen and hemoglobin. The experimen-
248-
Pflfigers Arch. 384 (1980)
Table 1. Influence of Pso on the estimated values of oxygen permeability
(o~Dc)and diffusion coefficient of hemoglobin
(DH)Pso
{Hb] = 0.224 kg/1
[Hb] = 0.152 kg/1
kPa (mm Hg)
aDc DH ~Dc
DH
10_x s mol
10_2~tmZ
10 15 mol
10_agm 2
s m Pa
ms
s m Pa
ms
0.06 (0.454)
15.6 • 3.4 %
2.43 • 6.9 %
18.0 _+ 3.3 %
3.99 _+ 6.5 %
1.57 (11.8)
15.4 +_ 3.5 %
2.73 • 6.2 %
17.7 _+ 3.2 %
4.51 _+ 5.3 %
4.82 (36.4)
14.9 _+ 3.4 %
3.47 • 4.3 %
17.1 +_ 2.7 %
5.79 _+ 3.4 %
t89
sec 60 50 &O 3O 2C 10~
ooj
I . j 10 20 30 /.,0 g % HbFig. 7. Experimental results of Klug et aL (1956) for 100 pm layers of
hemoglobin solution (1 g % = 10 g/l) using pure oxygen as oxygenat-
ing gas (P1 = 93.1 kPa or 700mmHg under their conditions),
ti/2= time needed to reach half saturation. Open circles are data points
obtained by assuming the deflection of the oximeter to be pro-
portional to the saturation increase. Crosses are data found in the
present study. Solid line is the regression curve fitting the data points
according to Klug et al. (1956). Broken line is the regression curve
corrected for the nonlinear relationship between oxygen saturation
and light intensity and therefore presents a better estimate of the
values of
qntal results o f Fig. 3 c o n f i r m the t h e o r e t i c a l a s s u m p t i o n s
a n d c o n c l u s i o n s in t h a t :
1. the curves at a n y one P1 in Fig. 3 c o i n c i d e for
l a y e r thicknesses in the r a n g e o f a b o u t 5 0 - 2 5 0 g m ,
w h i c h s u p p o r t s the a s s u m p t i o n o f c h e m i c a l e q u i l i b r i u m
in this r a n g e ;
2. the s a t u r a t i o n in the presence o f t w o diffusing
s u b s t a n c e s (02 a n d H b ) increases, u p to a h i g h e x t e n t o f
o x y g e n a t i o n , p r o p o r t i o n a l l y to the s q u a r e r o o t o f t i m e
as p r e d i c t e d b y the t h e o r y ;
3. the d e v i a t i o n o f the o x y g e n a t i o n p r o c e s s f r o m the
c o u r s e p r e d i c t e d b y the semi-infinite m o d e l o c c u r s a t
l o w e r values o f s a t u r a t i o n as P1 d e c r e a s e s ;
4. at P1 = 93.1 k P a ( 7 0 0 m m H g ) the o x y g e n a t i o n
p r o c e s s is t e r m i n a t e d a b r u p t l y , w h i c h evidences a s h a r p
o x y g e n a t i o n f r o n t w i t h i n the l a y e r a n d a p p r o a c h e s the
m o v i n g f r o n t m o d e l .
3 E 2"- 2 E :x. 1 {9 a 0 0 ~ • ---.--x..~....~ .1 . 2 ,3 [Hb] [ k g / I1Fig. 8. The diffusion coefficients of oxygen
D ccalculated from the
experimentally found permeability by using values for the solubility
as discussed in the text. They are compared with the curve (solid line)
suggested by Kreuzer (1970) for the values of Dc found in the
literature. The dots refer to the class A experiments, the crosses to the
class B experiments. Temperature is 25~
T h e d i f f u s i o n coefficient o f o x y g e n is d e r i v e d f r o m
e D c
(Fig. 5) after c a l c u l a t i n g c~ as d e s c r i b e d in M e t h -
ods. T h e values for the respective h e m o g l o b i n c o n c e n -
t r a t i o n s agree w i t h the curve p u b l i s h e d b y K r e u z e r
(1970) w h o c o m p i l e d p r e v i o u s d a t a f r o m the l i t e r a t u r e
(Fig. 8).
T h e v a l u e s o f the d i f f u s i o n coefficient o f h e m o -
g l o b i n D H f o u n d here are c o m p a r e d w i t h values p u b -
lished b y o t h e r a u t h o r s in Fig. 6. T h e d a t a r e p o r t e d b y
R i v e r o s - M o r e n o a n d W i t t e n b e r g (1972) a n d M o l l (1966)
were c o r r e c t e d to a t e m p e r a t u r e o f 25~
A t e m p e r a -
ture coefficient o f 3 % / ~
has b e e n u s e d w h i c h is
b e t w e e n the values o f 3.3 % in the r a n g e o f 2 5 - 3 7 ~
a n d 2 . 3 % in the r a n g e o f 1 5 - 2 5 ~
as r e p o r t e d b y
K e l l e r et al. (1971).
F r o m the w o r k o f M o l l (1966) o n l y the d a t a o f
s t r o m a - f r e e h e m o g l o b i n s o l u t i o n s are c o n s i d e r e d here.
M o l l (1966) e s t i m a t e d DH o f 3 % h e m o g l o b i n s o l u t i o n s
in l a y e r s w i t h a n d w i t h o u t filters, the l a t t e r p r o v i d i n g
values o f D H 1.37 times as large as the f o r m e r . H e n c e the
d a t a o f M o l l (1966) o n DH m e a s u r e d in filters were
J. A. E. Spaan et al. : Oxygenation of Layers of Hemoglobin Solution 249
multiplied by 1.37 to allow for the tortuosity of the
filters.
At a hemoglobin concentration of 0.025kg/1
20
(2.5 g ~ ) and lower there is good agreement between
four experimental studies, the three shown in Fig. 6
(approximately 7.8 9 10 -2 gm2/ms) and a study of
Lamm and Poison (t936) providing a value of 6.8
-~-15
9 10-2 gm2/ms at 20~ or, after temperature correction
.~.
also 7.8 9 10 -2 ~tmZ/ms at 25~
The value ofD~v = 5
E
9
lO-21am2/msf~
atl~
- 10
hemoglobin concentrations is significantly lower than
that found by the other authors. Also their values of DH
~'~
at higher hemoglobin concentrations are much lower
, -
than those of the other authors.
5
The diffusion coefficients of hemoglobin estimated
here from the oxygenation experiments are higher than
those found by other authors. At hemoglobin con-
00
centrations around 0.15 kg/1 (15 g ~ ) there is fair agree-
ment with the results o f Riveros-Moreno and
Wittenberg (1972) and Moll (1966), and around 0.29 kg/1
(29 g ~ ) with those of Keller et al. (1971). The difference
found in D~ by the various authors might well be a
result of their differing methods, e.g., the application of
6
porous materials to maintain a well defined layer of
hemoglobin solution. It should be noted that our esti-
mation of D H does not depend on any apparatus cor-
rection factor.
~E
Apart from influences of the experimental tech-
~ 4
niques on the values of D/j, differences in the values of
D H might be due to differences in the physical models in
which D H has been defined. In our physical model DH
can be regarded as a "tracer diffusion coefficient",
oxyhemoglobin molecules being considered as labeled
.,- 2
hemoglobin molecules (Spaan, 1973).
By means of laser scatter techniques the mutual
diffusion coefficient of hemoglobin has been estimated
by Alpert and Banks (1976). Their results agree with
those shown in Fig. 6 at hemoglobin concentrations
lower than 0.1 kg/1 (10 g ~). However, the dependence
on hemoglobin concentration is much less steep than
that shown in Fig. 6. At a hemoglobin concentration of
0.3 kg/1 (30 g ~ ) their D~, is 3 times larger than our
value9 Keller et al. (1971) experimentally studied both
the tracer and mutual diffusion coefficients of hemo-
globin and found no significant difference. It is diffi-
cult to interpret a large difference in numerical values
between the tracer and mutual diffusion coefficient9
Most recent data of Gros (1978) are, when corrected to
25~
a little lower than our curve, whereas the values
of Jones et al. (1978) are way above and rather resemble
those of Alpert and Banks (1976). More work may be
needed to resolve these recurring discrepancies.
There is fair agreement between the present results
and those of Kreuzer (1953) when plotting
t~/d 2
against
hemoglobin concentration for air and using the data for
flg o
22 /
/
o, ~ 7 %
h i t.1
.2
[Hb]
( k g / I )
i.3
fig b
r 7% a/!" 9f't"
3%o%
:/12~176
l~~ I i.1
O i I I0
,2
,3
[Hb]
( k g / I )
Fig. 9 a and b. Results of the preliminary experiments of the present
study (dots) compared with curves calculated from the class A and the
class B experiments, a presents results for P1 - 93.1 kPa
(700 mm Hg), b results for P1 = 18.6 kPa (140 mm Hg). The
numbers next to the groups of points show the percentage of
methemoglobin present in the respective experiments. Although
Smi
is
rather high, the agreement between the preliminary experiments and
the final experiments is fairly good
c~Dc
and D H according to the present investigation
(Spaan, 1976), though Kreuzer's data show a larger
scatter:
In the preliminary phase of this investigation oxy-
genation experiments were performed using hemoglo-
bin solutions with relatively high methemoglobin con-
centrations. In Fig. 9 the results o f these preliminary
250 Pflfigers Arch. 384 (1980)
experiments (dots) are compared with curves calcu-
lated from data of
eD c
and D H obtained from the
earlier and final experiments. The values of the extinc-
tions used were obtained from the wedge experiments
performed in the preliminary phase. The gases used in
the preliminary experiments contained 5 ~ CO>
Figure 9 a shows results for P~ = 93.1 kPa (700 mm Hg)
and Fig. 9b for Pa = 18.6 kPa (140 mmHg). Next to the
curves the respective percentage of methemoglobin is
indicated. There is remarkably good agreement be-
tween the calculated curves (which are based on
S m
= 0) and the experimental values. The relatively good
agreement between data points and calculated curves of
Fig. 9 may be explained by compensating effects;
increasing
Sm
decreases the oxygen binding capacity
and thus decreases
t~/d 2
whereas
D c
and
hD~
are lower
than when using a hemoglobin solution of the same
value of h but with S m = 0, leading to increased values
of
q/d z.
On the whole the presence of methemoglobin
thus does not seem to affect the results appreciably.
In the oxygenation times for 1/3 and 1/2 of full
deflection reported by Klug et al. (1956), the deflections
were assumed to be equal to the rise in saturation which
in fact they are not. The saturation belonging to 1/2 and
1/3 deflection can be calculated knowing the extinctions
of oxyhemoglobin and deoxyhemoglobin at the wave-
length of the light used. Assuming a linear relationship
between S and 1 / 7 the real times needed to reach
S = 1/2 and S = 1/3 can be estimated. The difference
between the values thus estimated and the original data
depends on the hemoglobin concentration as shown in
Fig. 7 for
ill2.
The influence of this correction is
considerable particularly at high hemoglobin con-
centrations; the corrections for
t~/3
are similar. The
present data (crosses) agree with the corrected measure-
ments of Klug et al. (1956) (broken line). After this
correction the value of the oxygen diffusion coefficient
for a 35.5 g ~ Hb solution becomes 6.4 . 10 -6 cm2/s
rather than 3.5 9 10 .6 cm2/s as reported by Klug et al.
(1956) and fits the curve of Kreuzer (1970) better
(Fig. 8).
A p p e n d i x
Influence of Bandwidth on Measurement of Oxygenation
Let I 0 (2) be the intensity distribution function of the
emitter and g()c) the sensitivity curve of the light
sensor. When applying the law of Lambert-Beer to an
infinitesimally small part A2 of the emission spectrum
the current generated by the sensor can be described by:
i= f
g(2)Io(2)e-~Ix)cdd2
(A1)
- - c o
where e ( 2 ) = extinction coefficient of solution as a
function of wavelength 2, c = concentration of absorb-
ing species.
An approximation of the solution ofeq. (A1) can be
obtained by introducing a function 6 (2) defined by the
relationship:
e 00 = e* + 6 ()~)
(A2)
where e* is a constant of arbitrary value. Substitution of
eq. (A2) into eq. (A1) results in:
2 = o o
i= e -~*ca f g(2)Io()Oe-a(~Ocdd2.
(A3)
, ~ = - - c o
Series expansion of the exponent in eq. (A3)
provides:
: = c o
i = e -~* c~
I g ('~) Io (,~) d,~
A ~ --CO
- e * c d J'f=co g(2)Io(2)~,) d2}.
(A4)
2 = - - c o
For small values of e*
cd
eq. (A4) can be approx-
imated by:
i
- - = e ~*ce ( A 5 )
io
2=w
where i 0 = . J"
g (2)I 0 (2)d2 is the "blank value" of
the system."= - co
In the red light c5 (2) is an even function of 2 for most
kinds of hemoglobin, and a value of e* may be chosen
to make the first-order expansion term in eq. (A4) equal
to zero. Hence the law of Lambert-Beer may still be
applied when introducing an adapted extinction coef-
ficient (e*) which may differ essentially fi'om the
extinction coefficient at the peak wavelength of the
emission spectrum. It may easily be seen that if different
light absorbing species are present in the solution the
summation rule of absorbing species, thus in fact eq.
(2), remains valid when using the adapted extinction
coefficient.
The validity of relationship (A5) has been shown to
hold in the region of layer thicknesses as used in the
oxygenation experiments by using the wedge method of
Spaan et al. (1977). The summation rule has been tested
by comparing the oxygen saturation as measured by the
wedge method with measurements of the oxygen con-
tent of the same samples by the Lex-Oz-Con (Spaan et
al., 1977).
The authors greatIy appreciate the generous cooperation and stimu- lation by Prof. Dr. P. C. Veenstra in this work. They wish to thank Jack Couwenberg and Mr. Severs for their contribution to the design and construction of the diffusion apparatus and of the oximeter respectively.
J. A. E. Spaan et al. : Oxygenation of Layers of Hemoglobin SoIution 25]
References
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Alpert, S. S., Banks, G.: The concentration dependence of the hemoglobin mutual diffusion coefficient. Biophys. Chem. 4, 2 8 7 - 2 9 6 (1976)
Assendelft, O. W. van: Spectrophotometry of haemoglobin derivatives. Assen, The Netherlands: Van Gorcum, Ltd. 1970 Christoforides, C., Hedley-Whyte, J.: Effect of temperature and
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Dijkhuizen, P., Buursma, A., Fongers, T. M. E., Gerding, A. M., Oeseburg, B., Zijlstra, W. G. : The oxygen binding capacity of human hemoglobin. Htifner's factor redetermined. Pfltigers Arch. 369, 223-231 (1977)
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