Mathematical analysis of oxygen transport to tissue

ERIC %.SALATHh, TSENG-CHAN WANG, Center for theApplication of Mathematics, L&&h Uaiwrsiity, Bethlehem Penmykxania18015

JOSEPH P. GROSS Department of Chemical Engineering U..wrsi@ of Arizona, Tucson, Arizona 85721 Received 21 Janwy

1980; mised 17 March 1980

ABsrRAcr

The mathe.maticalanalysis of steady state oxygen distribution in a Krogh cylinder is presented in this paper. P&urbation techniques are used to determine the effect of axial difksion when the diameter of the Krogh cylimier is small compared to its length. Separate asymptotic expansions -aredeveloped for the arterial end, the centid region, amd the venous end of the Krogh cylinder. Solutioae are obtained for each of these regions, and they are combined, using the methods of matched asymptotic expankons, to obtain a solution that is uniformly valid throughout the entire Mrogh cylinder. ‘Ihe accuracy of the perturbation analysis is examined by comparing the approximate solution with the exact solution for the case of a linear oxyhemoglobin dissociation relatiomhip. A simple ~analyticcriterion is derived for determining when axial diffusion is important and when it can be neglected.

INTR.ODUCTION

The determination of oxygen concentration profiles in a single capillary and in a surrounding (coaxial cylinder of tissue is a fundamental problem in the mathematical study of oxygen transport to tissue. Although the basic model was originally~ introduced by Krogh [7] for the study of oxygen distribution in the highly regular capillary bleds of skeletal muscle, it has also been applied to such irregular capillaty beds as cerebral gray matter [9]. In Krogh’s initial work only a highly simplified and rather elementary mathematical analysis of the model was presented. Since that time, however, considerable attention has been given to the development of more elaborate analyses. A summary of some of this work can be found in Middleman [8], and the proceedings of a recent symposium on oxylgen transport to tissue [2] contains numerous accounts of mathematical studies MATHAUWATICALBIOSCIENCES 5I.:89- 115 (1980) ollan(L,Inc., 1980 52 Vanderbilt Ave., New York, NY 10017

89 0025.5564/83/070089 + 27W2.25

9Q

E. P.

SALATHE, T.-C. WANG, AND J. F. GROSS

of the Krog]hcylinder, as &is model is now called. The very complex nature of the governingequations has always resulted in significant simplifications b&g ma& at the outset, so that the mathematics becomes tractable. Therefore, the analytical treatment is still far from complete. A mathematkal study of steady state oxygen dismbution in a Krogh cylinder is presented in this paper. The full equations are considered, including axial diffusion in the tissue and capillary, and an arbitrary oxyhemoglobin dissociation relationship is used. With the full oxyhemoglobin dissociation relationship, analytic solutions have been obtained previously only by neglecting axial diffusion in the capillary and in the tue. However, the solution obtained in an earlier paper on substrate concentrations surrounding single capillaries (&lathe and Wang [11]), equivalent to the present problem with a linear oxyhemoglobin dissociation relationship, clearly indicates that axial diffusion can have a significant effect on oxygen concentration profiles. It would therefore be very useful to have a simple analytic criterion, applicable to the full nonlinear problem, that would indicate when axial diffusion is important and when it can be neglected. Such a criterion is obtained in this paper. When the Krogh cylinder is long and slender, 0;’ when the ratio of axial to radial tissue diffusivity is small, the solution without axial diffusion provides the dominant effect, and axial diffusion may be included as a perturbation to that solution. It will be seen that such a perturbation is si~~~gular and that the methods of matched asymptotic expansions must be employed to pamplete the solution. The criterion for determining the significance of ax31 diffusion (#mergesfrom the application of this method. Separate solutions must be obtiined for the arterial end, the central region bounded away from the ends, and the venous end. These solutions may be joined in an appropriate manner using the methods of matched asymptotic expansions to obtain a uniformly valid asymptotic expansion applicable throughout the Krogh cylinder. In the earlier paper on substrate concentration profiles (Salathe and Wang [l ID, an exact solution to this problem was obtained for the case of a linear oxyhemoglobin dissociation relationship. The accuracy of the perturbation analysis presented here will be examined by comparing the perturbation solution of this linear problem with the exact solution. It will be seen that the two solutions are virtually identical, so that the validity of the approximate solution is established. FORMULATION OF THE PROBLEM The Krogh model, illustrated in Fig. 1, consists of a single capillary of length L and radius R, surrounded by a ::uncentric cylinder of tissue having radius R,. xygen transported by the blood diffuses from the capillary into

91

OXYCXN TRAlUSPOR‘I

FIG I, Geometry of the problem: a single @Nary surrouded by 8 ~oaxkl cylinder of tissue.

the tissue, where it is conmmed at a constant rate AZ.’Let z deaote distance akmg the capillary axis normalized with respect to L(0 ;I;;; z G It), and r distance normal to the axis normlized with respect to R,(O & r< 1). The oxygen concentration in the tissue, c,(r, z), satisfies the ec+.iom and boundary conditious

(2) ac -.=O, 32

z=o,1,

R
(3)

where c=c,/C, is nondimensionalized with respect to the oxygen concentration of the arterkl blood, CA. Here e==vD,/D, (R,/L), R=R./R,, MO= R fM/D&, an4 D,, D, are the radial and axial diffusivities of the tissue. ‘hling and Pittman [11]studied the oxygen sensitivity of smooth muscle by determining the relationship between thickness of carotid artery strips and the external PO2 ?hat limits contractile force. Tote minimum PO2 which could sustain full tension development continously decreased witlkdecreasing strip thickn~s, indicating that the apparent oxygen dependenct of the contra<:tileforce was due to difkion limitation. It may be concluded from these experiments that above approximately 2 mmHg, the oxygen consumption of the particular smooth muscle examined is not sensitive to oxygen concentrations.

92

E. P.

SALATHE,T.-C. WANG, AND

Y. F. GROSS

The oxygen content per unit vohune of blood is &+RS*(C,), where the first term represents the dissolved oxygen and the second term the oxygen bound to hemoglobin. The oxygen capacity of blood at 100% saturation is denoted by F, and S*(Cb) is the oxyhemoglobin dissociation relationship. This function may be approximated by the empirical formula

for suitable choice of the constants K aud n. The particular form of this relationship makes no difference to the present analysis, and Eq. (4) may be replaced by a tabular or graphical representation if desired. The oxygen concentration in the blo~I, Cb, varies with both radial md axial location within the capillary. HolMever, the radial variation may be neglected, as it is insignificant. A number of previous studies have considerably complicated the problem by attempting to consider a radial variation in oxygen c6acentration based on the aissumption of iaminar homogeneous fIow within the capillary. The passage of red blood cells single file through the capillary, separated from each other by a bolus of fluid, precludes such an assum@on. In addition, the recirculating flow within each bolus obwed by Brother0 and Burton [ 101results in convective mixing (Gross and Aroesty [Q, so that a uniform radial oxygen distribution is a more reasonabk assumption. Therefore the governing Iequation for the normal&d oxygen concentration within the capillary, C= C,/C,, is

O&2<

1.

(9

This equation states that the oxygen of the blood changes along the capillary as a rest& of axial diffusion within the capillary and radial diffusion into the surrounding tissue. IHere /3=q/2sD, LR, Q--R&/2 D, , W=E/CA, and S(C)=S*(C,,C), where q is the volume blood flow rate, and Db is the diffusivity of blood. The formulation is completed by specifying the oxygen concentration of” the blood at the arterial end, C(0)

-‘f

1

and by equating the oxygen concentration capillary wall,

(6) of the blood and tissue at the

METHOD OF SOLUTION It is clear from the governing equations tiat axial diffusion in the tissue and in the capillary become negligible in the limit e-Q. For capillary beds consisting of regular- parallel arrays of vessels, in which the distance separating the capillaries is small compared to their lengths, such as skeletal muscle, the Krogh cylinder is long and slender and R #/I, is small. However, since e= fD_i I:(R/L), the influence of axial diffusion depends not only on the geome%ry but also on the ratio of tissue diffusivities D,/D,. Axial diffusion in the capillary is small only if ESK 1. Therefore, for small e , 6= RD,/2 D, must be of order one or less. Since hemoglobin facilitates O2 diffusion, oxygen diXusivity in blood may be larger than in tissue. However, R = RJR tis generally small and should ensure that 6 is not a large number. 3%~ above equa$ions may be solved for small e by using perturbation methods. The dominant behavior is found by setting e=O, and corresponds to neglecting axial diffusion in the tissue and the capillary; tie higher order terms in the expansion give the effect of axial diffusion. Previously,, when a. nonlinear oxyhemoglobin dissociation relationship was used, analytic solutions were obtained only by neglecting the axial diffusion terms, This provides a sufficiently good approximation for extremely small capillary spacing, but significant errors will be introduced for less dense capillary networks. In this paper the effects of axial diffusion will be examined analytically, and it will be poesible to ascertain when these effects can be neglected and when they must be taken into account. &Gal diffusion in the capillary is generally less impbrtant than axial diffusion in the tissue, but it can also be included in the analysis without any difficulty. The solution for C(z) and c( r,zj will be obtained in the form of an asymptotic series, applicable for small e:

C(z)-C,(z)+&2C,(z)+

l

’

l

C(r,z)-C()(r,z)+E2C*(r,z)+-*-.

, (81

The leading terms, Co and co, satisfy Eqs. (f), (5) with ~0. Eqs. (l), (a), and (7) that

cop,

z)=Co(z)+

-‘n’-

R2

-z_

+blR

It fol1ows from

l

(9

Substituting this into Eq. \5) and using the btoundary condition (6), it follows that

E. P. SALKWE, T.-C. WANG,

94

AND J. F. GROSS

explititiy in terms Of Co. For ally choice of S( C,), a(Cts) can be computed, starting at Co = l,, Where2 = 0, and continuing until 1 is attained. It can be shown that this function is monlotornic,and so ghm, by inversion, the function Co(t). Sdutions of this type were obtained and dimmed by Kety 161. &uatiom’ for C,(z) and cl( r, t ) are dlbMmd by substituting the expan&on (8) into Eqs. (I>, (2)3 (5), (61, (?) and retaining terms of order e2. Using the expansion S(Co+e2C~)=S(C,)+e2!!~~(Co)C~, this gives Equation (10) gives

2

and hchdes the effect of axial diffusion. The solution to Eq. (I 1) satisfying the boundary conditions (IS), (14), is easily found to be

where C,(z) is already known from Eq., (10). Substituting this into Eq. (12) and integrating with respect to z gives

Using the boundary condition (14),, C,(O)==O,in Eq. (16) completes the solution for C,(z). It is not correct to use &is boundary condition, however, because the solution obtained in the form of the expansion given by Eq. (8) is not valid at the end z=O. The assumption that axial diffusion is a small effti that can be included as a perturbation is not correct at the ends of the Krogh cylinder, z = 0 and z = 1, It has not been possible in the present anal/ys!is to satisfy the boundary condition expressed by Eq. (3), and the s&.tkn C&)+e*C,(z) gives nonzero GJLygen flux thmgh the ends of the cylindrers. In the end regions the character of the solution changes radially from that obtained above, and radial ‘and axial diffu$on are of equal importance* In order to determine the unknown constant C,(O) and complete the solution for C,(z) [Eq. (16)] it is necessary to exanGne the solution in the region near z = 0. The procedure for doing this involves tie technique

OXYGEN TRANSPORT

95

of matched asymptotic expansions. A completely different expansion must be constructed for the solution in the neighborhood of z :=O, in which axial and radial diffusion are of equal importance, and which satisfies the boundary condition of zero flux through the end of the cylinder [Eq. (3)J. This expansion must be joined in an appropriate sentje to the solution already obvked for the region bounded away from the ~erncls. The matching of the two expansions will yield the required constant C,,(O).This procedure is described in lthe next section. THE SOLUTION NEAR z=O The solution obtained in the previous section is baseJ on thz assumption that as e-0 axial diffusgon becomes negligible compared ~9 2adia.l diffusion. This assumption is valid everywhere except near the ends, where no matter how small e is, there always exists a narrow region, or boundary layer, in which radial and axial diffusion are of equal importance. In these regions the character of the solution is entirely different from that alrea(dy obtained and satisfies the boundary condition of no flux throu& the icnds of the Krogh cylinder,. The solutions in these regions must be matched at the outer edge of the boundary layers to the solution obtained previously,. The boundary layer at the arterial end occupies a vanishingly small regicn about z :=O *& the limit e+O. Therefore it cannot be &scribed in terms of the physical variable z, since no matter how small z is, it is possible to choose e sufficiently small that 2 is not in the boundary layer. The appropriate bou.ndary layer variable is 2=2/e, which has the property z+O as e-+0 for fixed. 2. In terms of this variable the gwerninlg equations (1) and (5) are

where c(Z)=C(eZ) and E(r, Z)=c(r, eZ). In these variables, the radial and axial diffusaion terms iii Eq. (17) are of equal importance. The boundary conditions are 35

arI1” = ’ o

1

aE

az’

I

=0,

C(R,

2)-c(Z),

i?(O)=-1,

(19)

z-0

These must be supplemented by the requirements imposed by msktching this solution with t&e solution obtained earlier. In order to ~construct a solution to these equations in the i1orr.nof an asymptotic serks in E which joins properly to the previous solution,it is

E. P.

96

SALATHi!, T.-C.

WANG,

AND J. F. GROSS

w to examine the behavior of that solution as Z-SO. Since z+O is &Went to the limit e-O,2 fixed (recall that z = e Z), this can ‘bedone by rewriting the solutions C,,+e2CI, c(J+e2c, in terms of r and It, and expanding the result in a series in E. This gives

+eaZ

r2 T-lnr-~+lnR +e2p2+C,(o)-fCo”(o) I

)I

+O(e:),

m

where a=

(“d2fl)(R-1/R)

cc=-

=d

I+ NS’(1)

Ztcan be shown that a= q(O) and p-- &(O). the boundary layer expansion is of the form 54

+eaZ+e2P(

N(a2/2)S”(l) l+NS’(l)

’

It follows from Eq (20) that

Z),

)

+ln R +eJl(p, Z)+lt?2+(r, Z),

(21)

where equations and boundary conditions for P and JI are found by substituting this expansion into Eqs. (17), (M), and (19). (The corresponding equations foi + will be discussed later.) The result is

a it

==0, $=a&

210,

Red,

r=R, z’>O,

(22)

and F=p=2+

1 Ip[l+NS’(m

(23)

Comparing the expansions (20)2 and 1(21)~for c and c’, respectively, shows

OXYGEN TIUNWORT

97

that the two match if

With Eq. (22), this completes the specification of the problem for 4. Similarly, with P given by Eq. (231, comparing the expansions of C and C, Eqs. (20), and (21), respectively, gives

which provides the required car .stant C,(G). It is not necessary to solve Eqs. (22) and (24) for + in order to determine C,(O) from Eq. (25). Since the function I$ satisfies Laplace’s equation, the integral of its normal derivative over any closed surface S lying within the domain of definition of # must vankh. Choosing flor this surface the boundaries of the semiinfinite annular region in which $ is defined gives

Therefore, C’,(O)= -

M,( R- I/R)*

4p2 [ 1 +‘kS( 1)12

(26)

l

THE EFFECT OF AXIIQL DIFFUSION In order to determine the complete oxygen profile in the capillary and tissue, it is necessary to solve the boundary layer equations ob&ined in the previous section, as well as the correspon&ng boundary layer equations for the venous end. These boundary layer solutions and the solution in the central region bounded away from the ends can then be combined to provide a uniformly valid solution applicable throughout the Krogh cylinder. This will be done in subsequent sections. It will now he shown that the simple analysis presented here9 resulting in the determination of the unknown constant Ci(O& yields an effective criterion by which the importance of axial diffusion can be readily ascertained. Axial diffusion results in a more rapid decrease in capikry oxygen concentration at the arterial end, and oxygen is delivered to the tissue in excess of that necessary to meet its metabolic needs. This excess oxygen moves longitudinally in the tissue, resulting in a flatter capiiary profile downstream. The extent of the deviation of the oxygen profile from the solution without ax&l diffusion is

E. P.

98

!SALATH&T.-C.

WANG,

AND J. F. GROSS

reiated to the magnitude of e2CI(0). In terms of the dimensional Vtire%,

As t*C,(O)4, the effect of axial diffusion on the oxygen profile becomes negligible. For sn&l no-o values of e2C1(0) axial diffusion can be determined by means of the prturbation analysis. However, when e2C1(0) ti af the order crf I, axial diffusion and radial diffusion are equally important and the present analysis is not applicable. It is then necessary to to numerical methods. Figures 2-3 illustrate cqillary concentration profiles both with and without axiaI diffusion for a variety of physiological situations. The solutions with axial diffusion, C0+e2C,(Z), are not valid near the arterial and

1.0

0.8 --am_

o,c,

”

c (28 0.4

I)

0.2

0 0

0.2

0.4

0.6

0.8

1.0

2

Paa. 2.

The apillasy oxym wncentra~tissgC(z) for the data shown in Table 1, case values of tiubue !i!Nu&iviq: (“9wM!out 8.5X 1.7x IO3 pm2/sq (IIf?

c (:z) 0.5

CA = 0.0035

c

,;z)l’o Tz< 0.5

W c

-0

I

M = 0.0004 o--1cr

Q.a5

1.0

M=

O.OOf5

A. 6’

IFIG.3. The effect on the capowgen concentration of changes in various pmmeters: (a) increased oxygen content of the arterial blood, (b) decreased oxygen content of the arterhl blood, (c) decreased oxygen consumption rate, (d) increased oxygen conrsumptionrate; (I) without axial diffusion, (II) with axial diffusion.

ver~ous ends, z = 0, 1, and are therefore shown as broken lines in these r~qgions.The extent to which the solutiono with and without axial diffusion dltil~er,and therefore the importance of axial diffusion, is directly related to t!laemagnitude of e*Ci(O). The value of this constant is easily determined in e#ac:hfigure, since the solution with1axial Idiffusion has the value 1+e *C,(O) a#t;;;= 0. Figure 2 illustrates the solution without axial diffusion (curve I) and three solutions with axial diffusion for different values of tissue diffusivity D, =D, (curves IMV). The solution without axial diffusion is independent of the tissue diffusivity. The solutions correspond to the data shown in Table 1, case 1, except that curve II is for D,=O, = 850 pm*/sec and curve IV is for D,=O, = 2500 pm*/seq;. Clearly, both the importance of axial &fFusion and the absolute value of e*C,(O) increase as the diffusivity hc:rtSses.

loo

EmP. MLATHE,

T.-C. WANG, AND J. F. GROSS

Figure 3&b) shows the effect of changing the oxygen concentration of the arte&l blood. Roth these solutions a~renormalized with respect to the concentration given in ‘Table 1, case 1, C” = 0.0032 cm3 cm%lood, for comparison with Fig. 2. Comparing these sets of curves and curves I and 1% of Fig. 2 shows that as CA &creases, !&@)I and axial diffusion becomes less important. 3(c,d) illustrates the effect of changes in metabolic rate relative to 9 case! 1. Chuves 1 and III of Fig. 2 correspond to a e values WA in Fig. 3(c,d). These curves illustrate ter importance of axial diffusion at the arterial end, corresponding mcreased value of le2C,(0)i, when the metabolic rate is increased. the differences between the curves with and without axial diffuomparable in the midcapillary region. The reason can be seen from the form of the solution for C,(z), Eq. (16). of the factor [I +4W’(C&))], C,(z) decreases with ldistance z as slope of the oxyhemoglobin dissociatitm relationship increases. If enough is consumed in the tissue to move from the relatively flat portion of ociation curve to the very steep portion, there will be a significant in feW,(z)l from its initial value le*C,(O)l. Therefore, while f is mdk in Fig. 3(c) than in Fig. 3(d), the consumption rate is so l%g, 3(c) that there is not much decrease in le2CI(+)I along the y length. However, the larger (value of le2C,(0)l in Fig 3(d) deto much smaller values of Ie2Cl( z)l as the lower oxygen concentra0118 are attained. The nonlineatrity of the oxyhemoglobin dissociation relatior&ip plays an important role in cdetermking the difference between the sol~ti.on~ with and without axial Mfusion. Its effect, however, is to re axial diffusion below (that indicated by e2C1(0), so that this still remains a valuable guide in determking an uplper bound for the significance of axial diffusion. THE ARTERIAL BOUNDARY LAYER In the previous sections, solutions for the capillary and tissue oxygen ccncentration in the region bounded away from the arterial and venous ends have been obtained to order e2. The unknown constant C,(O) tl~~ appeared in these solutions was determined by constructing the boundary layer C~~W&I in the neighborhood of the arterial end, z = 0. la this section the asymptotic expansion near z’= 0 will be examined in maIre detail and solutior~ to the equations governing the oxygen concentration at the arterial =d will be obtained. From the expansions (20) it follows that, to order e2, the boundary layer expansions for the capillary and tissue concentrations are of the form given (21), where equatkms ad boundary conditions for P and + were obtained above, Substituting the expansion (21) into the boundary layer

101

equations (17)-(19) gives the following problem fe;>r9:

a+ -0

ar-’

!!L, a2

’

r=l,

z>o,

2~0,

Red,

(27)

The problem for \c,can be solved in terms of an eigenfunction expansion invoking composite BesseRfunctions: $-a&-4aa

g n-1

fm( r)eBhn*

(28)

A3,*2[_f”(l)]i--4h,’

The eigenfunctions f,( r) are defined in terms of the Bessel functions Jo and Yo bY

f,(r)=Yo(h,R)Ju(X,~)-J,(X,R)yo~~~~)? P) and the eigenvalues A, are the roots of ~~(Xa&t)r,(A)-

Y,(XR)J,(A)=O.

(30)

Here J,,,) Ymdenote the m th order Bessel functions of the first and second kind, respectivelly. It has been shown [ 131that Eq. (30) has no imaginary or repeated roots, and hss an infinite number of positive roots X, , n = 1,2,3,. 0+. These roots are not given in the literature for values of R of physiological interest, except for a few that were presented by ApelMat et al. [I]. For the present analysis additional eigenvalues have been calculated an/d bbulated 1121 &e functiion P( 2) describing the capillary oxygen concentration [cf. Eq. (21)] can be determined by substituting the solution for $(r, Z) into Eq. (23). This gives P(z)=pz2+cj(o)-

4M0(R- 1/R) g p2[l

+NS(l)f

where A.~(a2h2,R[Jb(1)12-4hlnR)-'.

n-1

A

ne-x,2

9

(31)

E

102

T.-C. WANG, AND J. F. GROSS

P.SATAm

flhc solution tc:bthe equations (27) for #Bis much more complex. Writing @(C z)+Pz’+G(o) --$

n-l

R2

r2

A&?‘~.+4

(

+lnr-T+lnR),

4M0~R-I/R)/j32[1+N~‘(l)]2,

(32) ’

it can be shown (Appendix I),

that

(33) with

E,(w=

This completes the determination of the boundary layer sol&kn to order e2

at the af&rial end. THE VEBJOInJS BOUNDARY LAYER The methods used in the previous section to obtain the arterial boundary layer solution can readily be extended to the venous end. In this case the boundaq layer variable is X=(1+/e, which has the property z+l as e+O for fixed X. In terms of this variable the governing equations (Mb),(B1) are

whercte(X)= C(1- eX) and Z(r, n’) =c( r, 1 - ex), and the ~undary coa_ &ions are

at

Jy,

at

E-0, p=Qq,

r==l, X=0,

X30 R
x>o.

(36)

103

OXYGEN TRANSPORT

Rewriting the solutions obtained above for c( r, z) and C(z) in terms of the new variable X and expanding the result in e gives the form of the venous boundary layer expansion and provides the matching conditions. It can be shown that

+elg(r,X)+e*~(r,X),

cm

l/R)/2/3[1+NS(C,(l))]= -C&(l) is analogous to a where &= -&(Rdeiined pre~o~ly. The rnat~~g conditions for P, 4, and 4 are found to be

where fi= -d*N,S”(C~(1))/2[1 +NS’(Co(l))]/e C”,(l)/2 is analogous to p. Substituting the expansions into Eqs. (35) and (36) yields an equation for P: as X+00,

-~~~~(x)[l+A~s~(c,(l))]+~~*x’s.(c,(l)))-~l,_n.

x>o.

The functions $ and 3 satisfy equations and boundary conditions completely analogous to $ and +. Their solutions are analogous to the solutions at the arterial end, and will not be given explicitly here (see [ 121for details). The function P(Z) for the capillary oxygen concentration in the arterial boundary layer satisfied an eqmtion similar to Eq. (39), and was solved subject to the initial condition P(O)= 0, since the oxygen concenrtraition at Z=O is known to be equal to 1 [cf. Eq. (6)]. The matching condition then served to determine C,(O). At the venous end, the reverse procedure applies. The constant C,( 1) in the matching condition, Eq. (38) 1$ is known, and provides the boundary condition for the solution to Fq. (39) for P(X). Solving Eq. (39) subject to the mat&ing condition Eq. (38) yields

where

d

E, P. S’UATHE,

104

T.-C. WANG, AND J. IF. GR.OSS

oxygen concentratio.n at X=4

or z= 1, is then given by Of course, if there is no axial diffusion of oxygen in the tha’cancentration at the veno~us ehd can be determined by a balance. Since such a mass balance is independent of axial in the tissue, it follows that at the venous end the capillary oxygen concentration is C*(l) and that &0)-O. It can be proven [I21 that if &=O, then the solution to Eq. (39) subject to the matching condition Eq, (38) es P(O)=O. Using similar methods for 6#0 yields the result

which vanishes when 6= 0, as required. A COMPOSITE SOLUTION

Three different expansions have been constructed to describe the oxygen distribution at the arterial end, at the venous end, and in the central region both ends. Outside of its individual domain of applicaon is invalid and the solution obtained from it is . However, the perturbation solution for the central portion has a Amman region of validity with each of the boundary layer solutions, and it is possible to construct a composite solution that is uniformly valid throughout the Krogh cylinder. This is done by adding together the solutions for all three regions, and subtracting any common p:ut of these solutions (see, for example, [3D. It can be seen that the common part of the arterial boundaq layer and the central region expansions is

ccp=1+eaz+q

pP+C,(O)]

and

_cq=l+2

A&

r*

(

+n~~+lnR

+e* /@+C,(O)-p [

R2 1

r*

(

+eaZ

R2

+.Ilr--2_+lnR

)I

for the capillary and tissue concentrations, respectively. Similarly, the common part of the venous boundary layer and central region expansions is

OXYGEN TRANSPORT

and

)I for the capillary and tissue concentrations, respectivel!y. The composite solutions for the capillary and tissue oxygen concen.trations, respectively, are therefore2 c(t)-C,(z)+e~[C,(z)+G(z, C(r,Z)uCO(r,Z)+eg(r,Z,X)

(431

X)],

where 4,X

G(Z,X)=-;

? (45)

g(r,

2, X)=

fnCr)

-4W g n-1 Xj,7T2[f,(l)l”_4Xn

(ap-hm;Z+&e-h~X)

9

(46)

and %

4M,(R-l/R) -m

= 82(1+NS(&(l))}’

’

2The rapid drop in oxygen conwntration near the arterial end results in wilsiderable variation in sl(C) through the art&al boundary layer. Since the oxyhenwglgbis dissociation relationship was expanded about z=O to obbairsthe boundary layer &&icn, S’(C) is replaced by F(l), which may rmt be a good approximation. (No such problem arises at the venous end, because of the rektively slow variation in C in that regic~~) It is possilble to obtain a modified composite solution that overcomes this difficulty. It can be shown that a uniformly valid expansion results if the factor [ 1+NS’( X:)]in Eq. (49 is replaced by 11+AW’(~(r:))J and if B”@‘) in Eq. (443 is multiplied by the factor [l +h’Y(CO(r))j/[ 1+ M’(l)]. Thk expansion uses the local value,, s”(C&)), &rough the arterial boundary layer.

E P. SALATH&

106

T.-C.

W'ANG, AND J. F. GROSS

A A#ODIFKATION OF THE ICROGH MODEL of the Krogh model is that oxygen delivered by a given c@llary lis consumed only in the cylinder of time surrounding it, and that this is the sole source of oxygen for the tissue cyhnder. This is a single capillary imbedcled in a large group of all originating from a given arteriole, as for example in skeletal muscle. ‘Ihe main defect occurs at the capillary ends, where there exists tissue not identified with any Krogh cylinder. Oxygen raazhes this tissue by km~tudincal diffusion, and a more appropriate model w4d be a Gogh q@inder with nonzero flux through the ends. The bow;. conditions alt z =Q and z = I, Eq. (3), are then replaced by “I&e basic assumption

wherefA=f,L/O,C,, and fo=Jz L/D,C' are nondimensional

fluxes. The cpaantities8; andf* are the constant oxygen fluxes through the arterial and venous ends, respectively. Their magnitudes are determined by the size and metabolic demands of the tissue not included in the Krogh cylinder. Except for the unknown mnstant C,(O), which was determined from the analysis of the: arterS boundary layer, there is no change in the solution for the region bounded away fr9m the ends. In the arterial boundary layer, #(r, 2) defii.ed in Eq. (21) must now be solved subject to the altered boundary condition Jl&,O)=J” [cf. Eq. (22)3]. This gives #(r,

Z)=aZ-4s(a-fA)

5 A:w*[ f"(l)]*-% fn(r)emhmZ

n-l

in place of the e&ier solution Eq. (28). The solution for +(t, Z), Eq. (32), is unchanged except for the replacement of 6 by 8( a -fA)/fi[l +NS’( l)]. The above modification of the solution for $(r, 2) results in a change in the capillary concentration:

which gives ‘it’)=

a-fA

2/3[1+NS’(l)]

l

107

OXYGEN TMNSP0RT

Since a < 0, the result of a positive f, (outflux) is an increase in e zC1(0), and therefore ani,increase in the importance of axial diffusion. Completely analogous alterations occur in the venous b,oundary layer solution (for details, see [ ilc!j). Uniformly valid composite solutions can be constructed,, and these are identical to the composite solutit>ns obtained earlier [Eqs. (43)~(46)] except that QLis replaced by a-j,,, and & by di-fu throughout. NUMERICAL RESULT’S AND DISCUSSION Oxygen concentration profiles in the capillary and tissue, corresponding to the data shown in Table 1, case 2, Gareillustrated in Fig. 4(a). The dash-dot line in FGg.4(a) shows the oxygen concentration along the length of the capillary when axial diffusion is neglected. The solution for the central region bounded away from the arterial and venous ends is given by the broken line. This solution is not valid at the ends and does ;aot assume

TABLE 1 Values of Parameters Used in Examples Parameter Arterial blood oxygen umcentration CA(cm30z/cm3 blood) Oxygen diffusivity D,, Dz 9& (d/W Oxygen capacity of the blood at 100%saturatim ~(cm30&m3blood) Oxygen consumption rate M(cm30z/cm3tissue set) capillarvl~gth L (crm) capillary radius R,(Ctm) Tissue radius R,(F) Volume blood flow rate dw3/=) Constant Kin oxyhemo-

Case1

case2

case3

chse4

3.2x 1O-3

2.8 x lO-3

28.8x 10”

3x 1o-3

1.7x lo3

1.7x lo3

1.3x 10’

2x 10’

0.204

0.204

0.204

0.204

8.5 x 1O-4

4X 1o-4

5x 1o-4

8x lO-4

200

400

200

150

3

3

3

25

30

30

30

25

1.13x lo4

5.66 x io’

1.13x lo4

‘7.85x 10’

8.55 x 10’

8.5§ x iOs

8.55 x lo5

-

2.0

2.0

2.0

-

gtobti &ssociation relationship constcuat n in oxyhemoglobin dissochtion rehtionship

(4

0.8

0.6 C(z)

--z--l .

0:4

0.6I

0.8 I

1.0 1

0.8

1.0

2

1.0

(b) i

‘\

‘\

0.8

O-6 c(r,z) 0.4

0.2

0

0.2

0.4

0.6 2

0.6

0 0.2

0

0.4

0.6

0.8

1.0

z

0.6 c(r,z)

0.4

0.2

0 0

0.2

0.6

0.4

0.8

1.0

2

Fro. 5. (a) Nomdized capillary oxygen concentration C(z) for the generalized Krogh model. (b) Normal&d tissue oxygen c~nce&ation, c(r, z), for the generalized Krogh modei as a function of axial position at a location midway between the Capillary wall and the outer edge of the ICroghcylinder. For tbc data iu Table 1, case 3. - - -without axial diffusiool, - - - - - with axial diffusion, composite soluthm. 109 l

l

l

110

E. P. SALATHE,

T.-C. WANG9 AND J. F. GROSS

the correct value at z ==0 and z = 1. The arterial boundary layer solution is shown in the figure hy the dotted line.3 This provides the correct profile aear z -0, but is invalid for larger values of t. The composite solution, which is uniformly valid throughout the length of the capillary, is given by the solid line. This solution was obtained neglecting axial diffusion in the capillary,and so has the same value at z= 1 as the solution without axial diffusion. The oxygen concentration in the tissue as a function of axial position is shown in Fig 4(b), corresponding to a location midway between the capillary and the outer edge of the Krogh cylinder (r=[R+ 1]/2). Again, the dashdc~itline is the solution without axial diffusion and the broken line is the solution for the region bounded away from the ends. Neither of these solutions satisfies the no flux condition at 2 = 0,l. The solid line represents the composite solution, and includes both the arterial and venous boundary layer solutions. Clearly ac/&=O at z =0, 1 for this solution. Figure 5 illustrates the solution for the modified Krogh model. In this case the data of Table 1, case 3, has been used, and an outflux of oxygen from either end qual to 10% of the total amount consumed in the Krogh cylinder assmed. The dash-dot lines correspond to the solution without axial diffusion, the broken line is the solution bounded away from the ends, and the solid line is the uniformly valid composite solution. Figure 5(a) shows the oxygen profile in the capillary. The composite solution attains a lower oxygen concentration at z= 1 than the solution without axial diffusion, since the former reflects the oxygen loss from the ends while the latter does not. ‘The tissue oxygen concentration is shown as a function of axial position at the mid cylinder section in Fig. S(b). The slope of the composite solution at z = 0,l reflects the outward flux of oxygen at these locations. The accuracy of the perturbation analyGs can be examined by applying it & the problem using a linear oxyhemoglobin dissociation relationship, and comparing the result with the exact sAution (Math& and Wang [l 1D. me linearized oxyhemoglobin dissociation relationship has the form S*(Ch)=SO+S,Cb. This cm be fitted approximately to the nonlinear relationship by choosing So =0.7 and &==80 (cm30&m3blood)‘? This reduces the nonlinear problem formulated here to the linear problem solved by SalaM ztnd Wang [III], except that q must be replaced by q[l +pS,] throughout. Figure 6 shows the exact and perturbation solution for the linear problem, corresponding to the data shown in Table 1, case 4. The two dash-dot 3’Thebox8ndacy

t3, in &er to o&

solution at the art&al ehclwas &ed one more term, to order more accuratesoIutiox~~ This extcnsio>lis outlinedin agpendixII. A

solution is not t solution to Q(

and the e venous

0.6

0

0.2

0.6

0.4

0.8

1.0

2

0.6

0

0.2

0.6

0.4

0.8

1.0

2

FIG. 6. Chnparison of the concentration profiles for the linearize4 problem usmg the perturbation solution and the exact solution, for the data shown in Table 1, case I: (a) capillary concentration, (b) tissue concentration at a location midway between the der. -.-a-•without capillary wall and the ou diffusion, - --.-_. --. composite solution, - ---e-e t solution. 111

112

E. P, SALXI’HE, T.-C. WANG, AND J. F. GROS2

liner in Fig, 6(a) give the ccapillaryoxygen concentration profile without axial diffusion, and the effect of axial diffusion in the central region ded away from the ends. The solid line represents the composite and the broken line tine exact solution. The close agreement the two is evideztt from the figure. Similar results are shown in Fig. 6@) for the oxygen concentration in the trissue. This figure illustratesI the &sue concentration as a function of axiaI position at a location midlway between the capillary ktnd the outer edge of the Krogh cylinder. The very close amem between the composite solution (solid line) and the exact shtioa (bmka fine) b qain evident. h each vz tissue concentration profties have been illustrated only for a radial location midway between the capiby and the outer edge of the Kmgh cyhier. For the (data used here, the profiles at other radial locations differ oniy slightly from the ones shown and therefore have not been ill&&d. The exCellent agreement between the exact solution and t;xleperturbation s&tion for the bear problem demonstrates the validity o$ the method of analysis developed in this paper. It may be inferred that the asymptotic analysb yiekis a suitable approximation for the nonlinear probkm, at least under some ran@ of conditions, A good indication of the accuracy of the approximate solution is obtained by o&n&g the extent to which the compo&e sokion first foiIows the boundary layer solution and then the central re@on soWion as it passes into their respective domains of validity, as illustrated in Fig 4(a), for exmple. A very large value of e2CI(0), or of the initial derivative e2C;(0), will result in a breakdown of the approximate solution. The first parameter is the difference between the solution without axial diffuskn and the solution for the central region at the location t ~0, and the second is the difference in their slope at z =0, as illustrated ixaFigs. 2,3. The demonstration of the valiclity of the analysis supports using the magnitude of e2Ct(0) as a criterion for determining the effect of axial diffusion. if this parameter is extremely small, axial diffusion can be ignofed. For moderate values of this parameter, the effect of axial diffusion can be taken into account using the present analysis. If 1e 2C, (0)) is large, it an be conch&d that axial diffusion is very important, the present analysis breaks down, and it is n-at-y to resort to elaborate numerical techniques to determine the concentration profiles.

To solve the equations (27) for cp(r, Z), the transformation (32) is introduced to define a new variable @(r, 2). This function satisfies the

113 equation

defined in the semiinfinite strip 2 > 0, R Q r ( 1, and the boundary con&tiOnS @=O,

r=R,

aa?

==a

r=l,

Z>O,

Z>O,

(52) (53)

The solution will be obtained as an eigenfunction expanSion of the form shown in Eq. (33), where the eigenfunctions j’ are defined in Eq. (29) and the functions Ek( 2) are to be determined. Using the properties of the eigenfunctions, it can be shown (see [ 121for details) that (56) where

Lk =2([ fk(l)]2-4/Af~2}~?

=-

It therefore follows that

Lk

where Eq. (51) was used to derive the second equality. Through integration by parts and the application of the bounldtary conditions (52) and (ST), this can be reduced to

since Q-0 aS z-+00 [Eq. (55)], it fOlhIWSthat &(z)+o aS z+oO. A second boundary condition for &( 2) is obtained from Eqs. (54) and (56).

E P.

SALA’l’E& T.-C. WANG,

AND 5. F. GROSS

solution to the second order or&ary differential equation for &(z), Eq. (57), subject to these boundary conditions yields the result given in Eq.

In this appendix the determination of the arterial boundary layer solutiog to order e3 will be briefly described. To this order, the boundary layer lution is given by [cf. (Zl)] C-1 +eaZ+e*.P(

Z)+e’Q(

Z),

(58)

(59) Matching conditions for Q and 0 are obtained by carrying out the eqxmof the solution in the region bounded away from the ends [Bq. (2011to order e’. This gives

as Z-+OO, where q= -aN[a*S”(1)+6pS”(l)]/[l +NS’(I)]=~C”(O)/6. Substituting the expansion (58) into the boundary layer equations gives

and the boundary condition Q(O)=O. Since P and + are known, the solution for Q is readily dekrmined. It can be shown that this solution satisfk the matching conditio* Eq. (60). Substituting the expansion (59) into the undary layer equations gives the following equation and boundary conditions for i9(r, 2): I a tar’z+

ae

al

I

r-1

ae

=o,

a*8 _. --a2:*

ae

’

R
Z>O,

e(R,Z)=Q(q. zz z_*=o

I

OXYGEN ~S~RT

115

Together with the matching condition (al), this provides a complete problem for the function O(r, 2). The solution can be obtained in a manner completely analogous to the method of solution of the problem for +( rp 2). This work was supported by N. I. H. Program Project Grant PO1HL-I7421 and NSF Grant No. ENG 77021542. REFERENCES 1 A. Apelblat, A. Katzir-Katchalsky, and A. Silberberg, A mathematical analysis of capillary-tissue fluid exchange, Biorhlw 11:l-49 (1974). 2 D. F. Bruley and H. I. Bicher, (Eds.), Owgen Tr-0~ ~0 Z&sue, in R-a in Experinaenta/ Medicine and Biology, Vol. 37B, Pfenum, New York, 1973. 3 J. J. Cole, Pertwhution Metho& in Applied Mathemutics, Blaisdell, Waltham, Mass., 1968. 4 B. R Duling and R N. Pittman.,0xygen tension: Dependent or independent variable in local control of blood flow?Fedhation Pm. 34:2012-2019 (1975). 5 Je F. Gross and J. Aroesty, ‘I&e mathematics of capitlary flow: A critical review, Biorko&y 9:225-264 (1972). 6 S. S. Kety, Determhm ts of tissue oxygen tension, Federation Proc. 16:666-670 (1957). 7 A. Km& The number and distribution of capillaries in mu&s with calculation.; of the oxygen pressure head necessaq for supplying the tissue, J. P&viol. 52:409-415 (1919). 8 S. Middleman, Traqport Phenomena in the Cardiowscular @stern, Wil+ntzfscience, New York, 1972. 9 E. Opitz and M. Schneider, The oxygen supply of the brain md the mechdsm of deficiency effec?s, Ergeb. Physiol. Biol. Chem. Expt. Pharmakol. 46:126-260 (19%). 10 J. Psothero and A. C. Bm& The physics of blood flow in capillaries, I, Bi&w. J. 1565-578 (l%l). 11 E. P. Salathh arid T. C. Wang, Substrate concentrations in tissue surrountig si@e capillaries, Math. Biosci. 49:2319-247 (1980). 12 T. C. Wang, Mathematical studies of oxygen transport to tissue, Ph.D. dissertation, I&gh Univ., Bethlehem, Pa. (1978). 13 G. N. Watson, A Tmtise on the Zheoty of BesseI Functions, 2nd ed., Cambridge, 1952.