Transfer of oxygen into haemoglobin solution

Transfer of oxygen into haemoglobin solution

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Spaan, J. A. E. (1973). Transfer of oxygen into haemoglobin solution. Pflügers Archiv : European Journal of Physiology, 342(4), 289-306. https://doi.org/10.1007/BF00586101

DOI:

10.1007/BF00586101

Document status and date: Published: 01/01/1973

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Pfliigers Arch. 342, 289--306 (1973) 9 by Springer-Verlag 1973

Transfer of Oxygen into Haemoglobin Solution

J. A. E. Spaan

Department of Mechanical Engineering, Section of Production Technology, University of Technology, Eindhoven, The Netherlands

Received April 9, 1973

Summary. It is shown that, when oxygen and haemoglobin are diffusing through

haemoglobin solutions, the concentration of total haemoglobin will be constant, independent of the reaction scheme of haemoglobin with oxygen.

Solutions of transport equations for the cases of oxygen uptake in a stationary and in a moving flat film of haemoglobin solution are given. The influence of the diffusion of haemoglobin is shown.

Advancing front equations, corrected for physically dissolved oxygen, are derived by formal integration of the transport equations. It is shown that ~hese derived formulas are good approximations of the solutions of the transport equations when the diffusion of haemoglobin is neglected.

Key words: Oxygen Transfer -- Haemoglobin -- Facilitated Diffusion -- Oxy-

genators.

A n u m b e r of Studies on the diffusion of oxygen in haemoglobin solu- tions and whole blood h a v e been published. Their p r i m a r y goal has been to describe the oxygen u p t a k e in the lungs a n d the oxygen release in the tissues. During t h e last 10 years new interest has arisen in connection with the problems in artificial gas exchangers.

I n oxygen transfer in haemoglobin systems there are three interacting p h e n o m e n a : convection, diffusion, and chemical reaction. F o r several geometric configurations and b o u n d a r y conditions numerical results of equations, describing oxygen transfer in various haemoglobin or blood systems, are available b u t almost always approximations were made, such as chemical equilibrium between oxygen and haemoglobin, neglect- ing the diffusion of haemoglobin, or assuming a simple one-step reaction for the oxygen-haemoglobin system. I n this p a p e r the influence of dif- fusion of haemoglobin is studied using an equilibrium relation between oxygen pressure a n d haemoglobin saturation in the case of:

1. non-stationary oxygen u p t a k e in a non-moving flat film,

2. s t a t i o n a r y oxygen u p t a k e in a moving flat film with parabolic velocity profile.

Comparison of c o m p u t e d numerical r e s u l t s , including diffusion of haemoglobin, with experimental results provides some indication con- eerning the influence of reaction velocity on the transfer process.

290 J . A . E . Spaan

Since the influence of haemoglobin diffusion on oxygen transport depends on the transfer of all (oxy-) haemoglobin species the differential equations used, describing oxygen concentration and haemoglobin satu- ration, are based on the transfer equations of all species involved.

1. Transfer Equations

Diffusion processes are normally described b y one or a set of differen- tial equation(s) holding in every point of the medium in which transfer occurs. Many solutions of this (these) equation(s) are possible but bound- a r y values for the particular problem determine the specific solution.

When oxygen passes through haemoglobin solutions it will react with haemoglobin to form oxyhaemoglobin. The transfer of all the species involved in the reaction can be described by:

a [x]

0t -- ~. grad [X] + div (Dx grad [X]) + R x (1)

where

[X] = the concentration of a certain species X,

R x = rate of production of species X b y chemical reaction,

D x = diffusion coefficient of X, = velocity of fluid.

[We will refer to an equation of t y p e (1) for species X as Eq. (X).] I n this paper we limit ourselves to the assumption t h a t in all situations the oxyhaemoglobin molecules remain in the four-heme form. Eq. (1) holds for oxygen of concentration C and for all haemoglobin species with concentration H~,~ where i = 0,1,2,3 or 4 indicates the number of 0 3 molecules bound to haemoglobin and ] distinguishes the various forms of oxyhaemoglobin with the same number of oxygen molecules.

Taking into account all forms of haemoglobin there is a relation between Rc and the various .RH, z, when R e and R~,,~ are the rates of production b y chemical reaction of C and Hi,j respectively:

4

R c + Z i Z RH,.~ = 0 . (2)

i = 1 i

t~or all possible forms of haemoglobin:

4

.Y, Z RH,.~ = 0 . (3)

i = 0 j

We also assume t h a t the diffusion coefficient is the same for all haemoglobin molecules. When defining b = ~ 2: Hr162 as the total haemo-

Transfer of Oxygen into Haemoglobin Solution 291

globin concentration (tetrameric) we now can derive a differential equation for b as a function of time and place. Adding all equations describing H,, I and using (1) we get:

~b

--~" grad b ~- div (D~ grad b) (4) St - - "

Because of Eq. (3) there is no reaction t e r m left. Eq. (4) shows t h a t in a system where 03 and haemoglobin species are moving and the men- tioned assumptions apply, the local total haemoglobin concentration is only affected b y gradients of b.

We consider here only situations where b is a uniform concentration a t the start. I n this case, if there is no change in b o u n d a r y values of b, no gradients of b will occur and b has a constant value throughout the whole process. The conclusion t h a t b is constant is required for the deri- vation of Eq. (7) below b u t also circumvents the need of proving it again for a certain specific problem as was the case hitherto (e.g. Kutchai, 1971a; Kreuzer, 1970; Zilversmit, 1965).

The fractional oxygen saturation of haemoglobin is defined as

4

S = (X i X H , , t ) / 4 b . (5)

i = 0

We can easily see t h a t the s u m m a t i o n

1 4

leads to

where

aS _~

0* - - - - v. grad S -t- div

(DH

grad S) -[- R s (7)

R s -~ ~,i XR~,.j : -- R c / 4 b . (8)

i = 1 i

Thus the saturation m a y be expressed b y a transfer equation as if S were a t r a n s p o r t e d species itseff.

Eq. (C) and Eq. (4 b S) can be added to give

( C + 4 b S ) - ~ ---~ g r a d ( C + 4 b S ) ~ d i v [ g r a d ( D c C - ] - 4 b D S ) ] . (9) Ot

I n addition to Eq. (C) and Eq. (S) we need equations for R v and R s depending on C and S. The formulation of these equations depends on the assumed reaction scheme for haemoglobin and oxygen.

The assumption of near equilibrium means t h a t the saturation curve m a y be used as a second relation required in addition to (9). We m a y

292 J.A.E. Spaan

qualitatively understand the meaning of this assumption from following considerations.

Generally we m a y state according to basic t h e o r y of reaction kinetics (Laidier, 1965) t h a t

- R c = / 1 c' - l~ ( l o )

where ]1 is a function of all terms H~,j of haemoglobin combining w i t h one or more molecules of oxygen (forward reaction) and ]2 is a function of all terms H~,I slowing down the overall rate (back reaction). ]1 and ]2 are always positive.

Defining C e as the concentration of oxygen for R e --- 0 (equilibrium) we get

Rv = (W -- C) h . (11)

Relation (10) holds when only one O 3 molecule is involved in each reac- tion step. I f there are more molecules involved per step then (10) wil be a polynome and Eq. (11) still holds as a first order approximation.

The assumption of near-equilibrium means t h a t C e - - C is smal compared to C e. I n this case a significant value of-Be is obtained ony when fl is v e r y large. Indeed ]1 m a y be large since it m a y contain reaction velocity constants of large numerical value.

2. Non.Steady State Oxygen Diffusion in a Non-Moving Film of Haemoglobin Solution; One-Dimensional Problem

We will mathematically describe the problem of a fiat thin layer of haemoglobin solution of thickness d. At t < 0 this film is in equilibrium with a gaseous atmosphere containing oxygen at a partial pressure P0. The oxygen concentration in the layer will be C o according to H e n r y ' s law

C O = cr Po (12)

where cr = the solubility in reel/l/ram Hg.

At time t = 0 the gaseous atmosphere changes abruptly to a Pc, of P r Oxygen will diffuse into the haemoglobin layer and react with haemo- globin to produce oxyhaemoglobin. I n this case Eq. (9) reduces to:

a (4 b S + C)/Ot = D e ~2C/aX2 -~- 4 b D ~ O2S/Ox 2. (13) The b o u n d a r y conditions are:

t < 0 P c , = Pc and C = Co in the entire layer,

Transfer of Oxygen into Haemoglobin Solution 293

a C / a x = 0 at x -~ d because no transfer of oxygen is occurring at this

surface. Together with the saturation curve as the relation between S and C, Eq. (13) with the boundary conditions can be solved. The standard oxygen dissociation curve at p H 7.4 and 25~ was used.

I n order to get a better insight into the influence of the various para- meters on the solutions of the equation, we will make all variables and parameters dimensionless. o * = v i o l b* -~ 4 b/C 1 y = x / d t* : t D v / d 2 D* ~- b* DH/Dc. (14)

Eq. (13) becomes with (14)

a (C* ~- b* S ) / a t = 02C*/Oy ~ + D*O~S/Oy ~ . (15)

The diffusion coefficient of haemoglobin, DH, has often been neglected hitherto because it is small compared with the diffusion coefficient of oxygen, D~. B u t on the other hand the molar haemoglobin concentration is usually much larger t h a n the oxygen concentration so t h a t the respec- tive fluxes of oxygen and haemoglobin m a y be of comparable size.

Eq. (15) is solved numerically following an adapted Crank-Nicholson difference scheme. Convergence of the method is proved both b y changing the number of grid points into which the film is divided and b y the number of time steps. Typical numerical results of profiles are shown in Figs. 1, 2 and 3. The boundary values are P0 ---- 0 and P1 : 0.9 arm according to the experiments of Klug et al. (1956) or P0 : 0 and P1 --~ 0.2 arm (air). F o r P1 : 0.9 there exists a sharp boundary between oxygen- ated and nonoxygenated haemoglobin. With P1 : 0.2 the diffusion of haemoglobin is relatively more important (see Fig. 3) and the oxygenation boundary is less sharp.

Comparison with Experimental Values. Klug et al. (1956) published

experiments on the same system as considered here. These authors determined the time for a thin haemoglobin layer t o be saturated to 1/3 and 1/2 as a function of hs concentration.

Their experimental values and our numerical results are compared in Fig. 4. We see t h a t the calculated results agree fairly well with the experiments when b is smaller than 30 g~ At higher haemoglobin con- centration the deviation between theory ~nd experiment c~nnot be explained b y existing uncertainty in diffusion coefficients. Finite reaction velocity in a n y ease slows down the oxygen uptake as compared to equi- librium situation. The discrepancy between numerical results and experi-

294 J. A. E. Spaan

b o,o o,1 o2 o,3 o,4 o,s o.s o,7 o.8 o,9 1,o

Y

:Fig. 1. a) C * as a function of y with t* as a parameter, b) S as a function of y with t*

as a parameter. - - - - D ~ = 0; DH = 0.4510 -G cm2/sec. [Hb4] = 10 g~ = 1.55 t . D

10 -a reel/l; Px ~ 0.9 arm; C x = 0.943 10 -a mol/1, t* = d--- ~ . (1) t* = 0.160; (2) t* = 0.544; (3) t* = 1.36.

These graphs are typical and do not differ in shape by changing b between 5 and 40 g~ The oxygenation front is less sharp when DH :fi 0. The lines indicated b y AF1, AE2 and A F s are the places of the oxygenation front calculated by the

advancing front hypothesis

m e n t s m i g h t b e c a u s e d b y finite r e a c t i o n v e l o c i t y , b y i n a c c u r a c y in t h e m e a s u r e m e n t s , or p e r h a p s b y a n a l t e r n a t i v e r e a c t i o n s c h e m e for t h e r e a c t i o n o f 0 2 w i t h h a e m o g l o b i n . I n F i g . 4 t h e t r i a n g l e s a r e t h e r e s u l t s c a l c u l a t e d b y K u t c h a i et al. (1971a). K u t c h M t a k e s i n t o a c c o u n t t h e r e a c t i o n v e l o c i t y b y a p p r o a c h i n g t h e 4 - s t e p r e a c t i o n b y a 1-step r e a c t i o n scheme. H i s r e s u l t s d e v i a t e f r o m

Transfer of Oxygen into Haemoglobin Solution 295

q

0,2 0,3

(3,0 0,1 0,4 0,5 0,6 0,7 0,8 0,9 1,0 Y IL

Fig.2. a) C* as a function of y with t* as a parameter, b) S as a function of y with t*

as a parameter. - - - - DH = 0; DH = 0.45 10 -6 em2/see. [Hb4] = 10 g~ = 1.55

10 -3 reel/l; P1 = 0.2 arm; C1 = 0.21 10 -~ reel/1. (1) t* = 0.538; (2) t* = 2.42; (3) t* = 5.64.

These graphs are typical and do not differ in shape by changing b between 5 and 40 g~ The oxygenation front is less sharp when DH =fi 0. The lines indicated by A ~ I , A~v 2 and A F 3 are the places of the oxygenation front calculated by the ad-

vancing front hypothesis

o u r s i n t h e w r o n g d i r e c t i o n b e c a u s e r e a c t i o n v e l o c i t y slows d o w n t h e o x y g e n u p t a k e a n d t h u s h i s v a l u e s f o r tl/2 a n d tl/3 s h o u l d lie a b o v e o u r c u r v e r a t h e r t h a n b e l o w . I n o u r o p i n i o n t h e g r i d size i n h i s c o m p u t e r p r o g r a m w a s t o o l a r g e . K u t c h a i c a l c u l a t e d t h e s a t u r a t i o n a n d o x y g e n c o n c e n t r a t i o n o n l y a t 10 p o i n t s o f t h e film. A s w e c a n s e e f r o m F i g . 1 t h e s a t u r a t i o n c h a n g e s o v e r 800/0 w i t h i n 1/20 p a r t o f t h e film.

296 g . A . E . Spaan / ~ ~ t ~ / ~ sl .4- 3- 2- 0,1 1,0 a

7-

~:~s,6.

~'"'"

J

2. t t* . Io b 1 I I I l I l~ I [ I L I I I t I I I I r 2 t I J E Fig. 3. Average saturation as a function of time. For S > 0.2 ~he solid lines are straight. - - - - D x = 0; - - D H =/= 0; a) C 1 = 0.943 10-3mol/1; b) 6~x = 0.21 10 ~mol/1. (1) b = 1 0 g ~ D ~ = 4 . 5 l0 -Tem2/see; (2) b = 3 0 g " / . D H = 0 . 9 4

I0-7 em~/see.

The influence of the diffusion of haemoglobin is larger if the 0~ gradients are smaller (C 1 smaller). The d o t e d lines are very well predicted by the advancing front hypo-

thesis

I n o u r p r o g r a m we p r o v e d also c o n v e r g e n c e w h e n t h e n u m b e r o f g r i d p o i n t s increases. W e n e e d e d u p t o a t l e a s t 80 s t e p s t o g e t a c c u r a t e results. T h e s a m e c r i t i c i s m h o l d s for Moll (1968) a l t h o u g h he u s e d ~ l o w e r pO 2 a t t h e b o u n d a r y a n d finite r e a c t i o n v e l o c i t y , so his s a t u r a t i o n profile in t h e film will b e m u c h s m o o t h e r . W e d i d n ' t use t h e s a m e b o u n d a r y con- d i t i o n s as Moll so t h a t we a r e n o t a b l e t o c o m p a r e his r e s u l t s w i t h ours. F u r t h e r m o r e we d o u b t w h e t h e r t h e r e a c t i o n Hba~ 4 03 c a n b e a p p r o x i - m a t e d b y a one s t e p r e a c t i o n s c h e m e o v e r t h e whole r a n g e o f 0 - - 1 0 0 ~ s a t u r a t i o n .

I t is i n t e r e s t i n g t o see t h a t a t h i g h e r h a e m o g l o b i n c o n c e n t r a t i o n s t h e difference b e t w e e n t h e c a l c u l a t e d r e s u l t s w i t h D ~ = 0 a n d DH --fi: 0 still r e m a i n s . T h i s is t o b e e x p e c t e d since t h e v a l u e s o f D * i n (15) a r e q u i t e i n d e p e n d e n t o f t h e h a e m o g l o b i n c o n c e n t r a t i o n in t h e r a n g e s t u d i e d h e r e ( T a b l e 1) e v e n t h o u g h t h e n u m e r i c a l v a l u e o f t h e d i f f u s i v i t y o f h a e m o - g l o b i n v a r i e s t e n - f o l d .

Transfer of Oxygen into Haemoglobin Solution 297 t l h 60-

~

/,,S

40-

pi-S."

30- . !~,

.<'5"J

20-

./.

~6 o z . . ~ / o - .~." " 0 10-

~

o

- ""s" " ~ " - ~

0

0 0 I I I I 10 20 30 40 Hb 4 in gr%

Fig.4. Computed results compared with experiments of Klug et al. (1956) and calcu- lations of Kutchai (1971a). tl/~. in seconds, o - data and regression line respectively from experiments by Klug et al. (1956); - - . - - theoretical curve when DH = 0;

--

-- theoretical curve where DH is taken from Table 1; A -- numerical results of Kutchai (1971 a)

Advancing Front Hypothesis. The a d v a n c i n g f r o n t hypothesis assumes a s h a r p b o u n d a r y between s a t u r a t e d a n d u n s a t u r a t e d haemoglobin, a linear [03] g r a d i e n t in t h e s a t u r a t e d layer, [03] --- C o a n d S --- S 1 in t h e u n s a t u r a t e d layer.

I f x / i s t h e distance o f t h e s a t u r a t i o n f r o n t t o t h e liquid-gas interface, a relationship can be f o u n d for x / a s a f u n c t i o n of time. F o r t h e average

oxygen

saturation

of

the haemoglobin

layer

one can derive:

]/

-

S(t)

---- 2 t D c (01-- Co) (1 - - S~) q- S~. (16) d 2 4b(1 - - 8 0

Marx et al. (1960) gave a solution o f Eq. (15) for D n = 0 b y means o f an analog c o m p u t e r a n d concluded in a qualitative w a y t h a t t h e a d v a n c i n g f r o n t assumptions are valid [Eq.(16)]. Q u a n t i t a t i v e conclusions, how- ever, c a n n o t b e d e d u c e d f r o m their report.

I f we assume t h a t D B = 0 a n d integrate Eq. (15) over t h e thickness of t h e film, we arrive a t :

i 1 y = l

= -~-]

(17)

0 0 y = 0

298 J. A. E. Spaan

Table 1. D* as a function of b

b g~ Dn" 107 em2/see Do" 105 em2/sec D*

10 4.5 1.6 0.185 20 2.05 1.2 0.224 25 1.3 1 0.222 30 0.94 0.83 0.224 35 0.65 0.67 0.222 40 0.44 0.58 0.199

10 gO/o ~ 1.55 10 -3 tool/l; C 1 ---- 0.943 10 -3 mol]l.

Du 4 b 4 b Dn

D * = b * b * = - - D* - - - -

D e ; 01 ; - - 01 D o "

The values of Do are taken from Kreuzer (1970) and those of D/t from data of Keller and Friedlander (1966) and compiled by Kreuzer (1970).

oxygenated haemoglobin (St) and using relations of S ( y ) and C ( y ) following the advancing front hypothesis provides:

] / - 2 t D c ( C,x - - Co) (1 - SO + (18)

S ( t ) = g2[db(1--,S',) +

~/~(O~--Oo)]

S~.

The difference between Eqs. (16) and (18) is t h a t in Eq. (18) 1/~ (01 _ Co ) accounts for the u p t a k e of u n b o u n d oxygen. Indeed Eq. (18) describes (within the accuracy of the numerical scheme used, which has an error of less t h a n 0.5 ~ average saturation) the results given in Fig. 3 adequately for the ease of DH = 0. The solution of Eq. (15) for ~q with D H = 0 is predicted v e r y well b y Eq. (18) over a wide range of C 1, b and Sl. The accuracy of the a p p r o x i m a t i o n depends on the value of C 1 and of the p r o d u c t 4b (1 - - S~). Our calculations show t h a t the accuracy is good when 4b (1 - - Si) ~ 10 -a reel/1 a n d C 1 ~ 0.410 -s reel/1 (P1 ~ 0.4 arm). The difference between (18) and (16) depends on the values of t h e factors mentioned above a n d m a y be between 1 a n d 10~ average saturation. Hill (1929) described the oxygenation process for Si ---- 0 b y solving the diffusion equation without chemical reaction for x smaller t h a n x]. A t x - ~ x ] the oxygen t r a n s p o r t is coupled with chemical reaction. Hill (1929) and also R o u g h t o n (1959) did the same for D H =/= 0 b y solving the same diffusion equation for the diffusion of haemoglobin a t x larger t h a n x]. I n fact b o t h authors included in their calculations the effect of the physically dissolved oxygen b u t arrived at equations different from (18). Hill and R o u g h t o n concluded t h a t their numerical results could be a p p r o x i m a t e d relatively well b y Eq. (16). T h e y defined a factor

Transfer of Oxygen into Haemoglobin Solution 299 h~ i n s u c h a m a n n e r t h a t Eq. (19a) p r o v i d e s t h e s a m e r e s u l t s as t h e i r n u m e r i c a l solutions. ]// 2tDoG~ (19a) ~- h~ de4b w h e r e i ~ 1 w h e n DH = 0 a n d i = 2 w h e n DH =fi O. h 1 is a f u n c t i o n o f C1/b a n d h 2 is a f u n c t i o n o f C~/b a n d DH/Do. R o u g h t o n (1959) p r e s e n t e d e x p l i c i t f o r m u l a s o f h 1 a n d h 2.

C o m p a r i n g E q . (19a) w i t h Eq. (18) shows t h a t w h e n E q . (18) p r o v i d e s a g o o d a p p r o x i m a t i o n o f t h e n u m e r i c a l r e s u l t s : / 1 (19b) h: = 1 + x/~ C1/4b " W e also c o m p u t e d h 1 a n d h 2 v a l u e s a n d c o m p a r e d t h e m w i t h d a t a p u b l i s h e d b y R o u g h t o n (1959) in T a b l e s 2 a n d 3. As far as h I is c o n c e r n e d t h e difference b e c o m e s l a r g e r for h i g h e r v a l u e s o f C1/b. T h e a p p r o x i m a - t i o n s o f H i l l a r e less s a t i s f a c t o r y in t h i s r a n g e . O u r d a t a for h~ in T a b l e 3 a r e c a l c u l a t e d f r o m ~ i g . 3. To b e a b l e t o c o m p a r e o u r v a l u e s w i t h t h e r e s u l t s o f R o u g h t o n t h e t a b l e f r o m R o u g h t o n (1959) was i n t e r p o l a t e d . H e r e we see t h a t t h e d e v i a t i o n is g e t t i n g l a r g e r w h e n Glib b e c o m e s s m a l l e r w h i c h is j u s t t h e o p p o s i t e o f h 1. I n t h i s p a p e r we will n o t f u r t h e r discuss t h e d e v i a t i o n b e t w e e n t h e d a t a o f R o n g h t o n a n d ours.

Table 2. Values of factor h 1 for various values of C1/b when D~ = 0. The first row

of figures are from Ronghton (1959), the second row of figures are calculated b y the author

C1/b 0.005 0.0202 0.0824 0.191 0.356 0.592 0.92

1 0.995 0.985 0.97 0.948 0,92 0.885

hi 0.998 0.995 0.980 0.955 0.921 0.878 0.827

T a b ~ 3. Values of factor h~ for various values of b/C 1 and DH/Dc. The first row of figures are a result of interpolation of a t a b ~ ~om Roughton (1959), the second row

of figures are calculated by the author from Fig. 3

b/01

6.5 19 29 89 0.0114 -- 1.08 - - 1.38 DH 1.07 1.27 Dc 0.0281 1.05 -- 1.28 - - 1.05 1.22

300 J . A . E . Spaan

K u t c h a i (1971b) concluded t h a t the advancing f r o n t hypothesis predicts v e r y well the results of Klug

et aL

(1956)." This conclusion is misleading. Recalculating the diffusion coefficients of oxygen from his results provides values of Do~ which are not in agreement with the values I ~ ' e u z e r published in his review of 1970 and K u t c h a i himself used in a previous p a p e r (1971 a).

3. Steady-State Oxygen Diffusion in a Moving Flat Film

of Haemoglobin Solution; Two-Dimensional Problem

W e now consider a haemoglobin solution streaming in a film along a flat plate or between two fiat membranes. I n b o t h cases there exists a laminair parabolic flow in the z direction for low Reynolds numbers. I n the case of the fiat plate, where film thickness ---- d, o x y g e n diffuses at x = 0 from the gas into the region where t h e velocity gradient is zero. W i t h the fiat membrane, where film thickness ~ - 2 d , oxygen diffuses f r o m b o t h sides into the region where the oxygen gradient is m a x i m u m . I n the case of two m e m b r a n e s we will consider only half of the film. Eq, (9) now reduces t o :

D a~o a~s

(20)

Vz(X) ~-~(4bS-4-C)= c - ~ - @ 4 b D H Ox a

with the b o u n d a r y conditions:

flat plate

fiat membrane

ao

- - - = O

x ~ d ;

--~x ~ 0

x ~ d ;

80az

x = 0 and z _ > 0 x = 0 and z > 0

C= 01

0=01

z < 0 a l l x

S=-St,

C = C o z < 0 a l l x S = S ~ , O : C O

(21)

Again we m a k e the problem dimensionless b y

C * = CIC1 z D c b* = 4 b / C , z * - - d~ vg y = x/d

D* = b* D~I/Do

V

vg

where vg -~ average velocity of the film

Transfer of Oxygen into Haemoglobin Solution 301 v* ~ (C* § b* S) - - a2c, ay~ 4- D * - _{~y2 9} (23) E q . (23) w i t h b o u n d a r y c o n d i t i o n s is n u m e r i c a l l y solved. 1 Define S(z*) - - f v*(z*, y) S(z*, y) dy . (24) o S o l u t i o n s o f S a r e s h o w n in Fig. 5. H e r e a g a i n t h e influence o f t h e diffusion o f Hb 4 is s m a l l d u e t o t h e h i g h o x y g e n g r a d i e n t s .

I n t h e case o f a flat film a p p r o x i m a t e s o l u t i o n s c a n be o b t a i n e d d i r e c t l y f r o m t h e a d v a n c i n g f r o n t h y p o t h e s i s (see also L i g h t f o o t , 1968; Buckles, 1966), or b y i n t e g r a t i n g t h e t r a n s p o r t e q u a t i o n s s i m i l a r t o E q . ( 1 7 ) . W e g e t for t h e flat p l a t e : N = S~ 4- (t - S~) - y p 3 4- ~ p (25) 1 Q)

4- u

(1 4 - Plate Membrane 1.0- ~ ~ . ~ o ,, ~ . . . . ? > . ~ . f . t . o O.8- . ~ 9 ~ x / ' 0.7 . /

o. ff

OA I ~ I 0 0.5 1.0 1.5 ~, 2.0 2.5 z:U

Fig. 5. Saturation as a function of z* in the case of fiat plate flow and flat membrane flow. DH = 0; . . . DH = 0.49 10 -s cm~/sec; o corrected advancing front equation; x uncorrected advancing front equation; Hb~ = 0.1125 10 ffi2 mo]/1;

C 1 = 0.943 10 -3 mol/l; C O = 0.16 10 -4 tool/l; St = 0.44.

There is fairly good agreement between the solid lines for DH = 0 and the circles for the corrected advancing front equation

where

302 J . A . E . Spaan

and for the fiat membrane (see also appendix 1) :

S ~ § (1 St) ( - - i 3 3 2

/

--- - - - 2 p § Y p / (26) z* : 3 [-- pa

(l/a2

§ Q/16) § pa (1/9 § Q/6)] p = x / / d (1 -- S,) 4b (27) Q - 1/2 (c1 - Co) "

The results are also shown in Fig. 5. For the case of the flat plate the advancing front gives slightly better results t h a n for the fiat membrane. I n this case again the advancing front equations corrected for the physically dissolved 02 provide better results in predicting ~ w h e n D r : 0. Lightfoot (1968) introduced, in his equations for the fiat membrane, b y intuition a factor i/2 (C 1 -- Co) to correct for the physically dissolved oxygen and arrived at the following equation for z* :

z* ---- 3 [-- p4 (1/16 § Q/16) § p a (1/6 § Q/6)]. (28)

Conclusions

We can conclude t h a t for the transfer of oxygen into haemoglobin solution, the oxygen saturation of haemoglobin obeys the same type of differential equation as does oxygen. The conditions are t h a t haemoglobin 9 remains tetrameric and t h a t the diffusion coefficient of haemoglobin does not change with the number of oxygen molecules combined with haemoglobin. Under these conditions and for the condition t h a t grad (b) ---- 0 at the boundaries, we proved t h a t the total haemoglobin concentration is constant.

We have shown t h a t the influence of the diffusion of haemoglobin has a considerable effect on the oxygenation process. This influence is larger when the oxygen partial pressure at the boundary, i.e. the [02] gradients in the layer, become smaller.

Comparison of the t h e o r y with experimental results does not lead to a clear-cut conclusion as to whether chemical reaction velocity plays a role or not. For haemoglobin concentrations lower t h a n 30 g~ there is fair agreement between t h e o r y and experiment. Reaction velocity, how- ever, slows down the process and therefore the t h e o r y would deviate more from the experimental results. At higher haemoglobin concen- trations the deviation is more pronounced. A change in Do2 of more t h a n 200/0 would be required to account for this deviation. Reaction velocity would approach t h e o r y and experiment b u t then its effect should be

Transfer of Oxygen into Haemoglobin Solution 303 l a r g e r a t h i g h h a e m o g l o b i n c o n c e n t r a t i o n s t h a n a t lower h a e m o g l o b i n c o n c e n t r a t i o n s . E x p e r i m e n t s w i t h l o w e r [02] g r a d i e n t s m i g h t g i v e b e t t e r i n s i g h t i n t o t h e role o f r e a c t i o n v e l o c i t y . F u r t h e r m o r e , t h e v a l i d i t y of t h e a d v a n c i n g f r o n t h y p o t h e s i s is t e s t e d w h e n DH is a s s u m e d t o b e zero. F o r m a l i n t e g r a t i o n o f t h e e q u a t i o n i n v o l v e d gives f o r m u l a s in w h i c h a t e r m a c c o u n t s for t h e p h y s i c a l l y d i s s o l v e d o x y g e n . T h e first q u e s t i o n , h o w e v e r , is w h i c h e q u a t i o n one a d o p t s t o d e s c r i b e t h e s y s t e m (e.g. DH ---- 0 or DH # 0) a n d t h e s e c o n d is w h i c h a p p r o x i m a t i o n is a p p l i e d t o t h e solution. W e s h o w e d t h e v a l i d i t y o f t h e a d v a n c i n g f r o n t h y p o t h e s i s in t h e case o f flat-film flow. F o r m u l a s (24) a n d (25) m a y b e v e r y useful in t h e case o f m e m b r a n e o x y g e n a t o r s a n d v e r t i c a l p l a t e o x y g e n a t o r s .

Symbols

cr solubility C e

b total concentration of haemoglobin DH

d film thickness

h l, h~ correction factors Dc

k~ reaction velocity constant of ith D x

reaction step (back reaction) D*

k~" as k~ (forward reaction) p x//d t time v ~ velocity vector v average velocity x position in x-direction

x/ position of oxygenation front

when fluid-gas interface at x = 0 R x

y x/d R~ z position in z-direction S b* dimensionless haemoglobin concentration t* dimensionless time S~ v* dimensionless velocity X z* dimensionless distance

Co,C i boundary conditions of [03] IX]

equilibrium concentration diffusion coefficient of haemoglobin

diffusion coefficient of oxygen diffusion coefficient of X dimensionless difhlsion coefficient

K, klTk~

Po~, P i boundary conditions of 02 partial pressure

Rc reaction velocity of oxygen

Rtt~, j reaction velocity of haemo- globin species

reaction velocity of species X saturation velocity

oxygen saturation of haemo- globin

average oxygen saturation initial saturation

species such as 02, Hbd, HbdO~, Hb~04, HbdOs, Hb~Os concentration of species X

21

A p p e n d i x I

Derivation of corrected advancing front equation for the fiat membrane case. Similar to Eq. (17) we get from (23):

1 1

fv*

+b,s) y

]dy.

(29)

0 0

304 J . A . E . S p a a n Since v* is only a function of y we get

1 1 1

,* (u* + b* 8) dv = ~--Vr du + D* J ~ dr. (30)

o o o

W e n o w are allowed to interchange t h e differential a n d integral signs. The r i g h t p a r t of Eq. (30) m a y be r e w r i t t e n leading to:

1 dz* (C*+b*S) v * d y : \ ~ - - / y = l - - \ - ~ - / y = o +D* ~ y = l o - D'~ [ a~ ~ (31) : \ ~ - / ~ , = o" Because of t h e b o u n d a r y conditions -~-v],=l - , ~ - / , = , = 0 .

D* [ aS ~ ~- 0 because no transfer of haemoglobin occurs t h r o u g h t h e

Also ~-~Y ]v = 0

m e m b r a n e - h a e m o g l o b i n interface. I n fact this last condition is in contradiction to t h e chemical equilibrium assumption. Since we compared the a d v a n c i n g f r o n t equations w i t h t h e o r y w h e n D* : 0 we will n o t go into details a b o u t this last con- dition. W h e n we assume a s h a r p b o r d e r between the oxygenated a n d non-oxy- g e n a t e d region a t position y ~ lo, we can rewrite Eq. (31) as:

[/

/

]

a { oc* ~ (3~)

dz* (C*+b*S) v*dy+ (C*+b*S) v*dy = - - \ Oy ]y=o" 1)

Because of tile a d v a n c i n g f r o n t hypothesis:

y ~ p S = I a n d C * : l - - ( ~ ) y (33)

y ~ p S = S~ a n d C* ~- Co*. (34)

So

aC* ~ i - - Co* (35)

ay I v = o = - p

Because of the parabolic velocity profile:

3 2

v * = 3 y - - ~ - y 9 (36)

AfLer s u b s t i t u t i o n of Eqs. (33, 34) a n d (35) into (32) we arrive a t forms we can inte- grate. Thus we get:

1 a

d [ ( ~ - p 2 - - ~ - l a ) [ ( 1 - - C o * ) + ( t - - ~ , ) b * ] - - ( l - - C o * ) ( l ~ 2 - - ~ - ' a )

Transfer of Oxygen into Haemoglobin Solution Since p is a function of z* we get:

3 2

[(3p _ ~_p ) [(l _Co,) ~ (l _ Sf) b, ] _ (l _ Co,) (2p _ 9 p ~ )] dp

dz*

1 - - C o *

P

We multiply Eq. (38) by p

.dz*/[1/2

(1 -- Co*)],

integrate subsequently and get:

z* = 3 [-- p~ (1/32 ~- Q/16) d~ p~ (1/9 d~ Q/6)]. Eq. (39) is the same as the second part of Eq. (26). Substitution of (33), (34) and (36) into Eq. (24) leads to:

p 1

3 y2 3 2 =

o p

-~ 24(1 - - ~ - p d - ~ - p ' ) = S ~ d - ( 1 3 2 1 _ Sl)( - y p 3 § ~ _ p 2 ) . 1 3 Eq. (40) is the same as the first part of Eq. (26).

305

(38)

(39)

(40)

References

Buckles, R. G.: An analysis of gas exchange in a membrane oxygenator. Ph.D. Thesis. Mass. Inst. Technology, Cambridge 1966

Hill, A. V. : The diffusion of oxygen and lactic acid through tissues. Prec. roy. Soc. B 104, 39--96 (1929)

Klug, A., Kreuzer, F., Roughton, F. J. W. : The diffusion of oxygen in concentrated haemoglobin solutions. Helv. physiol, pharmaco]. Aeta 14, 121--128 (1956) Kreuzer, F. : t~ber die Diffusion des Sauerstoffs in Blut. Helv. physiol, pharmaeol.

Aeta 9, 379--387 (1951)

Kreuzer, F. : Facilitated diffusion of oxygen and its possible significance; a review. Respir. Physiol. 9, 1--30 (1970)

Kreuzer, F., Hoofd, L. J. C.: Facilitated diffusion of oxygen in the presence of hemoglobin. Respir. Physiol. 8, 280--302 (1970)

Kutchai, H. : 02 uptake by 100 ix layers of hemoglobin solution: theory vs experi- ment. Respir. Physiol. 11, 378--383 (1971a)

Kutchai, H. : Wider applicability for Hill's advancing front theory of oxygen uptake. J. appl. Physiol. 81, 302--304 (1971b)

Kutehai, H., Jaequez, J.A., Mather, F . J . : Non-equi]ibrinm facilitated oxygen transport in haemoglobin soIugion. Biophys. J. 10, 38--54 (1970)

Laidier, K. G. : Chemical Kinetics, 2nd edition. New York: Mc Graw-Hfll 1965 Lemberg, R., Legge, J . W . : Hematin Compounds and Bile Pigments, p. 263.

London: Interseience Publishers Inc. 1949

Lightfoot, E. N. : Low-order approximations for membrane blood oxygenators. J.A.I.Ch.E. 14, 669--670 (1968)

306 J . A . E . Spaan

Longmuir, I. S., l%oughton, F. J. W. : The diffusion coefficients of carbon monoxide and nitrogen in haemoglobin solutions. J. Physiol. (Lond.) 118, 264--275 (1952) Marx, T. I., Snyder, W. E., St. John, A. W., Moeller, C. E. : Diffusion of oxygen

into a film of whole blood. J. appl. Physiol. 15, 1124--1130 (1960)

Moll, W. : The influence of haemoglobin diffusion on oxygen uptake and release by red cells. Respir. Physiol. 6, 1--15 (i968)

Pmughton, F. J . W . : Diffusion and simultaneous chemical reaction velocity in haemoglobin solutions and red cell suspensions. Progr. Biophys. 9, 56--104 (1959)

Zilversmit, D. B. : Oxygen-hemoglobin systems: a model for facilitated membranous transport. Science 149, 874--876 (i965)

Dr. J. A. E. Spaan

Dept. of Mechanical Engineering Section of Production Technology University of Technology Eindhoven/The Netherlands